Polar Code Based Federated Learning: Convergence Analysis and Resource Allocation

# Convergence Analysis and Resource Allocation
Source: [https://arxiv.org/html/2608.13961](https://arxiv.org/html/2608.13961)
## Polar Code Based Federated Learning: Convergence Analysis and Resource AllocationThanks:This work is partially supported by the National Natural Science Foundation of China under Grants6236114685362361146853, and6237112962371129, and the Research Fund of the National Mobile Communications Research Laboratory, Southeast University, under Grant 2026A05\.Thanks:Han Xiao and Wei Kang is with the School of Information Science and Engineering, Southeast University, Nanjing 211189, China \(e\-mail:hanxiao@seu\.edu\.cn; wkang@seu\.edu\.cn\)Thanks:Nan Liu are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 211189, China \(e\-mail: nanliu@seu\.edu\.cn\)

###### Abstract

Federated learning \(FL\) enables collaborative model training across distributed devices without sharing raw data; however, it faces significant communication bottlenecks and channel impairments in practice\. Conventional network layer treatments either idealize the channel as error free or apply equal error protection \(EEP\) to transmitted model updates, failing to account for the inherently unequal importance of quantization bits within a single local model\. To address this limitation, we propose a cross layer polar code based FL scheme that leverages the unequal error protection \(UEP\) property of polar codes under finite block lengths\. Specifically, the proposed design selectively protects more significant quantization bits, thereby mitigating the detrimental effects of channel noise\. We further provide a rigorous convergence analysis of the proposed scheme, deriving an upper bound on the convergence gap, which we then jointly optimize over the number of quantization bits and the polar code block length across all training iterations\. Experimental results demonstrate that both constant and variable block length configurations of our polar code based scheme consistently achieve substantial performance gains over uncoded and LDPC\-based EEP benchmarks, with the advantage becoming increasingly pronounced as the channel quality deteriorating\. These findings confirm the efficacy of our cross\-layer design in enhancing FL robustness and efficiency under realistic channel conditions\.

## IIntroduction

Conventional machine learning relies on a centralized paradigm, wherein training data is stored and processed in a data center or cloud infrastructure\[[1](https://arxiv.org/html/2608.13961#bib.bib1)\]\. In practice, however, privacy regulations and communication resource limitations often prevent users from directly transmitting their raw data to the central server\[[2](https://arxiv.org/html/2608.13961#bib.bib2)\]\. Benefiting from the rapidly increasing computing and storage capabilities of mobile devices, distributed learning, which enables local model training on individually collected data, has recently gained prominence\[[3](https://arxiv.org/html/2608.13961#bib.bib3)\]\. Federated learning \(FL\), one of the most promising distributed learning frameworks, allows users to collaboratively train a shared model without exposing their private datasets\. This approach offers substantial advantages in preserving data privacy and achieving low latency communication, as highlighted in recent work\[[4](https://arxiv.org/html/2608.13961#bib.bib4)\],\[[5](https://arxiv.org/html/2608.13961#bib.bib5)\]\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/FL1.png)Fig\. 1:Federated learning system\.Most existing studies on federated learning restrict their analysis to the network layer by assuming that each local model update is encapsulated within a single packet for transmission\. Specifically, the majority of these works, e\.g\.,\[[6](https://arxiv.org/html/2608.13961#bib.bib6)\],\[[7](https://arxiv.org/html/2608.13961#bib.bib7)\],\[[8](https://arxiv.org/html/2608.13961#bib.bib8)\],\[[9](https://arxiv.org/html/2608.13961#bib.bib9)\],\[[10](https://arxiv.org/html/2608.13961#bib.bib10)\],\[[11](https://arxiv.org/html/2608.13961#bib.bib11)\],\[[12](https://arxiv.org/html/2608.13961#bib.bib12)\],\[[13](https://arxiv.org/html/2608.13961#bib.bib13)\],\[[14](https://arxiv.org/html/2608.13961#bib.bib14)\],\[[15](https://arxiv.org/html/2608.13961#bib.bib15)\], disregards the inherently noisy nature of channels by assuming a rate limited yet error free channel\. Other works, such as\[[16](https://arxiv.org/html/2608.13961#bib.bib16)\],\[[17](https://arxiv.org/html/2608.13961#bib.bib17)\],\[[18](https://arxiv.org/html/2608.13961#bib.bib18)\],\[[19](https://arxiv.org/html/2608.13961#bib.bib19)\],\[[20](https://arxiv.org/html/2608.13961#bib.bib20)\],\[[21](https://arxiv.org/html/2608.13961#bib.bib21)\],\[[22](https://arxiv.org/html/2608.13961#bib.bib22)\],\[[23](https://arxiv.org/html/2608.13961#bib.bib23)\], explicitly consider the noisy transmissions and treat the packet error probability as as a tunable physical layer parameter, influenced by factors including channel conditions, bandwidth allocation, power control, and client selection, etc\., to directly optimize the overall convergence performance\.

Despite their diversity, the above network layer treatments suffer from two fundamental flaws\. The first one is latency: the error free channel assumption is theoretically justified only under infinite latency constraints, as per information theory\[[24](https://arxiv.org/html/2608.13961#bib.bib24)\]\. In practice, some works, e\.g\.\[[25](https://arxiv.org/html/2608.13961#bib.bib25)\], employ an acknowledgment\-and\-retransmission mechanism, which may incur unbounded delays on a small subset of users, causing the straggling effect that disrupts synchronized global aggregation\. The second flaw lies in the separation principle: at the network layer, compression and channel coding are treated as independent building blocks, with the physical\-layer implementation details largely abstracted away\. While this separation is valid for a sequence of sources according to information theory\[[24](https://arxiv.org/html/2608.13961#bib.bib24)\], it breaks down in FL, where each local model, whether scalar or vector, constitutes a single source symbol\. During scalar quantization, each bit corresponds to a distinct quantization level; yet conventional channel coding provides equal error protection \(EEP\) uniformly across all bits\. This design is suboptimal, as the quantized bits of a local model exhibit inherently uneven importance, thereby necessitating unequal error protection \(UEP\)\.

To overcome these limitations, this paper studies the FL problem from a cross\-layer design perspective and proposes a polar code based FL scheme\. By exploiting finite block length polarization, we intrinsically realize UEP to protect the quantized model bits according to their significance\. We further derive an upper bound on the convergence gap of the proposed scheme and optimize this bound over the number of quantization bits and the polar code block length across training iterations\. Our approach offers two key advantages\. First, the inherent UEP property of polar codes provides stronger protection for the more important quantization bits compared with traditional EEP channel coding schemes\. Second, our optimization reveals that allocating greater channel resources, specifically, longer polar code block lengths, to the later stages of training effectively suppresses the cumulative errors introduced by quantization, channel noise, and other impairments\. Numerical experiments demonstrate that our proposed scheme achieves significant performance gains over both uncoded and LDPC\-based EEP benchmarks\.

## IISystem Model

We consider a federated learning system, see Fig\.[1](https://arxiv.org/html/2608.13961#S1.F1), consisting ofMMusers, each indexed ask∈ℳ=\{1,2,…,M\}k\\in\\mathcal\{M\}=\\\{1,2,\\ldots,M\\\}and a server\. The task of the federated learning system is to minimize the following functionF⁡\(𝒘\)F\(\\boldsymbol\{w\}\)

F⁡\(𝒘\)≜1M​∑k=1MFk​\(𝒘\),\\displaystyle F\(\\boldsymbol\{w\}\)\\triangleq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}F\_\{k\}\(\\boldsymbol\{w\}\),\(1\)whereFk​\(𝒘\)F\_\{k\}\(\\boldsymbol\{w\}\)is the strongly\-convex local loss function for thekk\-th user, and𝒘∈ℝd\\boldsymbol\{w\}\\in\\mathbb\{R\}^\{d\}denotes thedd\-dimensional model parameter vector\. Every user has its own training data setDkD\_\{k\}with size\|Dk\|\|D\_\{k\}\|, and the data sets for different users are disjoint, i\.e\.,Dk∩Dj=∅D\_\{k\}\\cap D\_\{j\}=\\emptyset, fork≠jk\\neq j\. The overall data set for all users is denoted as𝒟=⋃k∈ℳ𝒟k\\mathcal\{D\}=\\bigcup\_\{k\\in\\mathcal\{M\}\}\\mathcal\{D\}\_\{k\}, while the size of all training data is\|D\|=∑k=1M\|Dk\|\|D\|=\\sum\_\{k=1\}^\{M\}\|D\_\{k\}\|\.

Let𝝃k\\boldsymbol\{\\xi\}\_\{k\}be the mini\-batch of thekk\-th user with sizebb, which is sampled from the datasetDkD\_\{k\}independently\. We denote the local empirical loss function with respect to mini\-batch samples𝝃k\\boldsymbol\{\\xi\}\_\{k\}as

Fk​\(𝒘,𝝃k\)=1b​∑i=1bf⁡\(𝒘,ξk​i\)\.\\displaystyle F\_\{k\}\\left\(\\boldsymbol\{w\},\\boldsymbol\{\\xi\}\_\{k\}\\right\)=\\frac\{1\}\{b\}\\sum\_\{i=1\}^\{b\}f\\left\(\\boldsymbol\{w\},\\xi\_\{ki\}\\right\)\.\(2\)whereξk​i\{\\xi\}\_\{ki\}is the i\-th sample in the mini\-batch𝝃k\\boldsymbol\{\\xi\}\_\{k\}, andf⁡\(𝒘,ξk​i\)f\\left\(\\boldsymbol\{w\},\\xi\_\{ki\}\\right\)is the loss function with respect toξk​i\{\\xi\}\_\{ki\}\. Each sampleξk​i\{\\xi\}\_\{ki\}consists of a pair\(xk​i,yk​i\)\(x\_\{ki\},y\_\{ki\}\), wherexk​ix\_\{ki\}is the feature andyk​iy\_\{ki\}is the label\.

Let𝒘∗\\boldsymbol\{w\}^\{\*\}be the optimal model weight vector, and letF∗F^\{\*\}andFk∗F\_\{k\}^\{\*\}be the minimum values ofFFandFkF\_\{k\}, respectively\. Then,Γ=F∗−1M​∑k=1MFk∗\\Gamma=F^\{\*\}\-\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}F\_\{k\}^\{\*\}represents the degree of data heterogeneity\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/FL2.png)Fig\. 2:Flowchart of the learning procedure in a single round\.The system diagram for the FL system studied in this paper is depicted in Fig\.[2](https://arxiv.org/html/2608.13961#S2.F2)\. In each round of the iteration, the system will conduct the following operations:

Broadcasting:Due to the limit of communication resources, not all the clients can participate in each learning round\. At the beginning of the\(t\+1\)\(t\+1\)\-th round, the central server randomly selectKKusers out ofMMusers uniformly\. The set of selected users in the\(t\+1\)\(t\+1\)\-th round is denoted by𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}, with\|St\+1\|=K\|S\_\{t\+1\}\|=K\. In the\(t\+1\)\(t\+1\)\-th round, the central server broadcasts the aggregated model parameter vector from thett\-th round,𝒘t\\boldsymbol\{w\}\_\{t\}, to the selected users in the set𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}\.

Local Model Updating:Each user in the set𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}conducts a mini\-batch stochastic gradient descent \(SGD\) method with the received global model𝒘t\\boldsymbol\{w\}\_\{t\}\. The gradient𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}is computed as

𝒈t\+1k=∇𝒘tFk​\(𝒘t,𝝃t\+1k\),\\displaystyle\\boldsymbol\{g\}\_\{t\+1\}^\{k\}=\\nabla\_\{\\boldsymbol\{w\}\_\{t\}\}F\_\{k\}\\left\(\\boldsymbol\{w\}\_\{t\},\\boldsymbol\{\\xi\}\_\{t\+1\}^\{k\}\\right\),\(3\)which can be used to update the model parameter vector locally as follows

𝒘t\+1k=𝒘t−ηt\+1​𝒈t\+1k,\\displaystyle\\boldsymbol\{w\}\_\{t\+1\}^\{k\}=\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\(4\)However, here we will transmit the gradient𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}instead of the model parameter vector𝒘t\+1k\\boldsymbol\{w\}\_\{t\+1\}^\{k\}through the channel\[[26](https://arxiv.org/html/2608.13961#bib.bib26)\]\. We note that𝒈t\+1k∈ℝd\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\in\\mathbb\{R\}^\{d\}is add\-dimensional vector and we denote thejj\-th component of𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}as𝒈t\+1,jk\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\.

Binary Erasure Channel:Each user in the set𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}uses a binary erasure channel with erasure probabilityϵ\\epsilon\(BEC\(ϵ\\epsilon\)\) to transmit the local gradient𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}to the server\. Assume that the input of BEC\(ϵ\\epsilon\) isXX, and the output of the BEC\(ϵ\\epsilon\) isYY\. The BEC\(ϵ\\epsilon\) is characterized as

Y=\{X,w\.p\.​1−ϵE,w\.p\.​ϵ,\\displaystyle Y=\\left\\\{\\begin\{array\}\[\]\{ll\}X,&\\text\{ w\.p\. \}1\-\\epsilon\\\\ E,&\\text\{ w\.p\. \}\\epsilon\\end\{array\}\\right\.,whereEErepresents the erasure occurred in the channel\. We denoteW:𝒳→𝒴W:\\mathcal\{X\}\\rightarrow\\mathcal\{Y\}as the BEC\(ϵ\\epsilon\) with the binary input alphabet𝒳=\{0,1\}\\mathcal\{X\}=\\\{0,1\\\}, the ternary output alphabet𝒴=\{0,1,E\}\\mathcal\{Y\}=\\\{0,1,E\\\}and transition probabilitiesW⁡\(y\|x\),x∈𝒳,y∈𝒴W\(y\|x\),x\\in\\mathcal\{X\},y\\in\\mathcal\{Y\}\. We assume that BEC\(ϵ\\epsilon\)WWis usedNNtimes consecutively and denote theNNmemoryless uses of the channel asWN:𝒳N→𝒴NW^\{N\}:\\mathcal\{X\}^\{N\}\\rightarrow\\mathcal\{Y\}^\{N\}withWN​\(y1N\|x1N\)=∏i=1NW⁡\(yi\|xi\)W^\{N\}\\left\(y\_\{1\}^\{N\}\|x\_\{1\}^\{N\}\\right\)=\\prod\_\{i=1\}^\{N\}W\\left\(y\_\{i\}\|x\_\{i\}\\right\)\. To deliver the gradient𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}throughNNuser of BEC\(ϵ\\epsilon\), we define encoder and decoder

ψ:ℝd↦𝒳N\\displaystyle\\psi:\\mathbb\{R\}^\{d\}\\mapsto\\mathcal\{X\}^\{N\}\(7\)φ:𝒴N↦ℝd\\displaystyle\\varphi:\\mathcal\{Y\}^\{N\}\\mapsto\\mathbb\{R\}^\{d\}\(8\)More specifically, the channel inputXn=ψ⁡\(𝒈t\+1k\)X^\{n\}=\\psi\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)and the reconstruction at the channel output is𝒈~t\+1k=φ⁡\(Yn\)\\tilde\{\\boldsymbol\{g\}\}\_\{t\+1\}^\{k\}=\\varphi\(Y^\{n\}\)\.

Aggregation:After reconstructing all the gradients of the users in the set𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}, the server aggregates the gradients and updates the global model parameter vector𝒘t\+1\\boldsymbol\{w\}\_\{t\+1\}as follows

𝒘t\+1=𝒘t−ηt\+1K​∑k∈𝒮t\+1𝒈~t\+1k\.\\displaystyle\\boldsymbol\{w\}\_\{t\+1\}=\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{K\}\\sum\_\{k\\in\\mathcal\{S\}\_\{t\+1\}\}\\tilde\{\\boldsymbol\{g\}\}\_\{t\+1\}^\{k\}\.\(9\)whereηt\+1\{\\eta\}\_\{t\+1\}is called step size or learning rate\.

## IIIProposed polar code based FL scheme

Existing works typically adhere to the separation principle, wherein the encoding and decoding blocks are decomposed into two distinct functions: quantization and channel coding\[[27](https://arxiv.org/html/2608.13961#bib.bib27)\]\. In this section, we propose a novel scheme comprising a joint design of unbiased quantization and polar codes\. Specifically, we first employ unbiased quantization to reduce the communication overhead\. Next, we adopt polar codes as the channel coding method to realize unequal error protection \(UEP\)\. Finally, we coordinate the compression and channel coding by assigning UEP to the quantization bits according to their relative significance\.

### III\-AQuantization

As mentioned above, to ease the burden of communication resources, the local gradients𝑺t\+1\\boldsymbol\{S\}\_\{t\+1\}of thekk\-th user will be converted into a quantized version, which is denoted asQ⁡\(𝒈t\+1k\)\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\. We assume that the gradient updates are bounded in the range\[Bm​i​n,Bm​a​x\]\[B\_\{min\},B\_\{max\}\]\. We assignnnquantization bits to transfer the real values into the discrete values represented by a binary sequence of lengthnn\. The quantization levels\{s0,s1,⋯,s2n−1\}\\left\\\{s\_\{0\},s\_\{1\},\\cdots,s\_\{2^\{n\}\-1\}\\right\\\}are uniformly placed between the upper bound and lower bound, i\.e\.,\[Bm​i​n,Bm​a​x\]\[B\_\{min\},B\_\{max\}\]is equally divided into2n−12^\{n\}\-1intervals where the width of each interval is

Δ=Bm​a​x−Bm​i​n2n−1\.\\displaystyle\\Delta=\\frac\{B\_\{max\}\-B\_\{min\}\}\{2^\{n\}\-1\}\.\(10\)For theii\-th interval\[si,si\+1\]\[s\_\{i\},s\_\{i\+1\}\], we note that

si=Bm​i​n\+i×Δ,i=0,⋯,2n−1\.\\displaystyle s\_\{i\}=B\_\{min\}\+i\\times\\Delta,\\quad i=0,\\cdots,2^\{n\}\-1\.\(11\)
So if the j\-th component of the d\-dimensional local update𝒈t\+1,jk∈\[si,si\+1\]\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\in\[s\_\{i\},s\_\{i\+1\}\], it will be quantized to

Q⁡\(𝒈t\+1,jk\)=\{si,w\.p\.​si\+1−𝒈t\+1,jksi\+1−sisi\+1,w\.p\.​𝒈t\+1,jk−sisi\+1−si\\displaystyle Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)=\\left\\\{\\begin\{array\}\[\]\{ll\}s\_\{i\},&\\text\{ w\.p\. \}\\frac\{s\_\{i\+1\}\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\}\{s\_\{i\+1\}\-s\_\{i\}\}\\\\ s\_\{i\+1\},&\\text\{ w\.p\. \}\\frac\{\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\-s\_\{i\}\}\{s\_\{i\+1\}\-s\_\{i\}\}\\end\{array\}\\right\.
Furthermore, thedd\-dimensional vector𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\}will be quantized separately in each component, i\.e\.,

Q⁡\(𝒈t\+1k\)=\(Q​\(𝒈t\+1,1k\)T,…,Q​\(𝒈t\+1,dk\)T\)T,∀k,t\.\\displaystyle Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)=\\left\(Q\(\\boldsymbol\{g\}\_\{t\+1,1\}^\{k\}\)^\{T\},\\dots,Q\(\\boldsymbol\{g\}\_\{t\+1,d\}^\{k\}\)^\{T\}\\right\)^\{T\},\\quad\\forall k,t\.\(14\)
The above quantization is unbiased as indicated in the following lemma\[[18](https://arxiv.org/html/2608.13961#bib.bib18)\]\.

###### Lemma 1

The stochastic quantization methodQ⁡\(⋅\)Q\(\\cdot\)is unbiased, i\.e\.,

𝖤⁡\[Q⁡\(𝒈t\+1k\)\|𝒈t\+1k\]=𝒈t\+1k\.\\displaystyle\\mathsf\{E\}\[Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\]=\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\.\(15\)

As the result of the quantization, each component of the gradientQ⁡\(𝒈t\+1,jk\)Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)is mapped into a binary sequence of lengthnn, denoted asBt\+1,j,lkB\_\{t\+1,j,l\}^\{k\}wherel=1,…,nl=1,\\dots,n\. The quantization interval represented by theii\-th bit is2i−1​Δ2^\{i\-1\}\\Delta\. We note that the quantization interval represented by different bits are not equal and therefore it requires the unequal error protection \(UEP\) property from channel coding\.

### III\-BPolar codes and Bhattacharyya parameters

Polar codes is a proven capacity achieving channel coding scheme\[[28](https://arxiv.org/html/2608.13961#bib.bib28)\]\. With block length going to infinity, the synthesized channels polarize to either error free or totally blocked\. However, in the case of finite block length, the synthesized channels polarize mildly, i\.e\., the qualities of some synthesized channel improve and some worsen, not reaching the0−10\-1extremes\.

To quantify this diverse performance of synthesized channels, we consider the Bhattacharyya parameterZ⁡\(W\)Z\(W\)defined as follows

Z⁡\(W\)≜∑y∈𝒴W⁡\(y\|0\)​W​\(y\|1\)\.\\displaystyle Z\(W\)\\triangleq\\sum\_\{y\\in\\mathcal\{Y\}\}\\sqrt\{W\(y\|0\)W\(y\|1\)\}\.\(16\)
With successive cancellation decoder being used, the Bhattacharyya parameterZ⁡\(WN\(i\)\)Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)is shown\[[28](https://arxiv.org/html/2608.13961#bib.bib28)\]as an upper bound of the probability of the bit error for theii\-th synthesized channel in the polar code, i\.e\.,

P⁡\(ℰi\)≤Z⁡\(WN\(i\)\),\\displaystyle P\\left\(\\mathcal\{E\}\_\{i\}\\right\)\\leq Z\\left\(W\_\{N\}^\{\(i\)\}\\right\),\(17\)
It is also shown\[[28](https://arxiv.org/html/2608.13961#bib.bib28)\]that in BECs, the Bhattacharyya parametersZ⁡\(WN\(i\)\)Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)can be calculated analytically in a recursive way as

Z⁡\(WN\(2​j−1\)\)\\displaystyle Z\\left\(W\_\{N\}^\{\(2j\-1\)\}\\right\)=2​Z​\(WN/2\(j\)\)−Z​\(WN/2\(j\)\)2,\\displaystyle=2Z\\left\(W\_\{N/2\}^\{\(j\)\}\\right\)\-Z\\left\(W\_\{N/2\}^\{\(j\)\}\\right\)^\{2\},\(18\)Z⁡\(WN\(2​j\)\)\\displaystyle Z\\left\(W\_\{N\}^\{\(2j\)\}\\right\)=Z​\(WN/2\(j\)\)2,\\displaystyle=Z\\left\(W\_\{N/2\}^\{\(j\)\}\\right\)^\{2\},\(19\)withZ⁡\(W1\(1\)\)=ϵZ\\left\(W\_\{1\}^\{\(1\)\}\\right\)=\\epsilon\.

We rearrange the labeling of synthesized channels in the polar code according to their Bhattacharyya parameters such that ifi<ji<j, we have

Z⁡\(WN\(i\)\)≤Z⁡\(WN\(j\)\)\\displaystyle Z\(W\_\{N\}^\{\(i\)\}\)\{\\leq\}Z\(W\_\{N\}^\{\(j\)\}\)\(20\)The ascending order shows that polar code in the finite block length possesses the unequal error protection \(UEP\) property, which is required from the quantization\.

### III\-CJoint design of quantization and channel coding

The quantized gradientQ⁡\(𝒈t\+1k\)Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)includesddcomponents, i\.e\.,Q⁡\(𝒈t\+1,jk\)Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)forj=1,…,dj=1,\\dots,d\. Each component is represented by a binary sequence of lengthnn, i\.e\.,\{Bt\+1,j,1k,…,Bt\+1,j,nk\}\\\{B\_\{t\+1,j,1\}^\{k\},\\dots,B\_\{t\+1,j,n\}^\{k\}\\\}\. We have

Q\\displaystyle Q\(𝒈t\+1,jk\)=\\displaystyle\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)=Δ⁡\(Bt\+1,j,nk​2n−1\+…\+Bt\+1,j,lk​2l−1\+…\+Bt\+1,j,1k\)\.\\displaystyle\\Delta\(B\_\{t\+1,j,n\}^\{k\}2^\{n\-1\}\+\\ldots\+B\_\{t\+1,j,l\}^\{k\}2^\{l\-1\}\+\\ldots\+B\_\{t\+1,j,1\}^\{k\}\)\.\(21\)As we mentioned at the end of the subsection of quantization, the quantization interval represented by different bits are not equal, therefore, we use the UEP property of the channel coding to protect the quantization bits\. More specifically, we feedBt\+1,j,lkB\_\{t\+1,j,l\}^\{k\}into thell\-th synthesized channelWN\(l\)W\_\{N\}^\{\(l\)\}in the polar code\. For the channel, where the label is larger thannn, we input a fixed bit, which is referred as frozen bit\. The output of the synthesized channelWN\(l\)W\_\{N\}^\{\(l\)\}is denoted asB^t\+1,j,lk\\hat\{B\}\_\{t\+1,j,l\}^\{k\}forl=1,…,nl=1,\\dots,n\. The probability of error ofWN\(l\)W\_\{N\}^\{\(l\)\}is denoted asPlP\_\{l\}and due to the symmetry of the channel and polar code, we have

Pl\\displaystyle P\_\{l\}=𝖯𝗋⁡\(B^t\+1,j,lk≠Bt\+1,j,lk\|Bt\+1,j,lk\)≤Z⁡\(WN\(l\)\)\.\\displaystyle=\\mathsf\{Pr\}\(\\hat\{B\}\_\{t\+1,j,l\}^\{k\}\\neq B\_\{t\+1,j,l\}^\{k\}\|B\_\{t\+1,j,l\}^\{k\}\)\\leq Z\(W\_\{N\}^\{\(l\)\}\)\.\(22\)
Before reconstruction, we shift the channel output byZlZ\_\{l\}to compensate the decoding error as follows

B¯t\+1,j,lk\\displaystyle\\bar\{B\}\_\{t\+1,j,l\}^\{k\}=B^t\+1,j,lk\+\(1−2​B^t\+1,j,lk\)​Zl\\displaystyle=\\hat\{B\}\_\{t\+1,j,l\}^\{k\}\+\(1\-2\\hat\{B\}\_\{t\+1,j,l\}^\{k\}\)Z\_\{l\}=\{1−Zlif​B^t\+1,j,lk=1Zlif​B^t\+1,j,lk=0\\displaystyle=\\left\\\{\\begin\{array\}\[\]\{ll\}1\-Z\_\{l\}&\\text\{if \}\\hat\{B\}\_\{t\+1,j,l\}^\{k\}=1\\\\ Z\_\{l\}&\\text\{if \}\\hat\{B\}\_\{t\+1,j,l\}^\{k\}=0\\end\{array\}\\right\.
This is a soft\-output method to compensate the bias introduced by the channel\.ZlZ\_\{l\}is an estimated value ofPlP\_\{l\}based on experimental experience which should satisfyPl≤Zl≤Z⁡\(WN\(l\)\)P\_\{l\}\\leq Z\_\{l\}\\leq Z\(W\_\{N\}^\{\(l\)\}\)\. Here we simply takeZl=Z⁡\(WN\(l\)\)Z\_\{l\}=Z\(W\_\{N\}^\{\(l\)\}\)\.

We apply the reconstruction procedure of the quantization and obtain

Q~\\displaystyle\\tilde\{Q\}\(𝒈t\+1,jk\)=\\displaystyle\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)=Δ⁡\(B¯t\+1,j,nk​2n−1\+…\+B¯t\+1,j,lk​2l−1\+…\+B¯t\+1,j,1k\)\.\\displaystyle\\Delta\(\\bar\{B\}\_\{t\+1,j,n\}^\{k\}2^\{n\-1\}\+\\ldots\+\\bar\{B\}\_\{t\+1,j,l\}^\{k\}2^\{l\-1\}\+\\ldots\+\\bar\{B\}\_\{t\+1,j,1\}^\{k\}\)\.\(25\)The difference betweenQ⁡\(𝒈t\+1,jk\)Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)andQ~​\(𝒈t\+1,jk\)\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)is given by

Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)=Δ​∑l=1n\(B¯t\+1,j,lk−Bt\+1,j,lk\)​2l−1\\displaystyle\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)=\\Delta\\sum\_\{l=1\}^\{n\}\(\\bar\{B\}\_\{t\+1,j,l\}^\{k\}\-B\_\{t\+1,j,l\}^\{k\}\)2^\{l\-1\}\(26\)

In this section, we conduct a convergence analysis of the proposed SGD algorithm and quantitatively show how the client scheduling, quantization, polar codes over BEC jointly affect the convergence performance of federated learning\. To facilitate the analysis, we make the following assumptions, which are widely taken in the literature\[[6](https://arxiv.org/html/2608.13961#bib.bib6)\]\[[11](https://arxiv.org/html/2608.13961#bib.bib11)\]\[[29](https://arxiv.org/html/2608.13961#bib.bib29)\]\[[30](https://arxiv.org/html/2608.13961#bib.bib30)\]\[[31](https://arxiv.org/html/2608.13961#bib.bib31)\]\.

###### Assumption 1

Each local loss functionFk​\(𝐰\)F\_\{k\}\(\\boldsymbol\{w\}\)isLL\-smooth, i\.e\.,

Fk\(𝒗\)≤Fk\(𝒘\)\+\(𝒗−𝒘\)T∇Fk\(𝒘\)\+L2∥𝒗−𝒘∥2,\\displaystyle F\_\{k\}\(\\boldsymbol\{v\}\)\\leq F\_\{k\}\(\\boldsymbol\{w\}\)\+\(\\boldsymbol\{v\}\-\\boldsymbol\{w\}\)^\{T\}\\nabla F\_\{k\}\(\\boldsymbol\{w\}\)\+\\frac\{L\}\{2\}\\\|\\boldsymbol\{v\}\-\\boldsymbol\{w\}\\\|^\{2\},\(27\)for any𝐰\\boldsymbol\{w\}and𝐯\\boldsymbol\{v\}\.

###### Assumption 2

Each local loss functionFk​\(𝐰\)F\_\{k\}\(\\boldsymbol\{w\}\)isμ\\mu\-strongly convex, i\.e\.,

Fk\(𝒗\)≥Fk\(𝒘\)\+\(𝒗−𝒘\)T∇Fk\(𝒘\)\+μ2∥𝒗−𝒘∥2,\\displaystyle F\_\{k\}\(\\boldsymbol\{v\}\)\\geq F\_\{k\}\(\\boldsymbol\{w\}\)\+\(\\boldsymbol\{v\}\-\\boldsymbol\{w\}\)^\{T\}\\nabla F\_\{k\}\(\\boldsymbol\{w\}\)\+\\frac\{\{\\mu\}\}\{2\}\\\|\\boldsymbol\{v\}\-\\boldsymbol\{w\}\\\|^\{2\},\(28\)for any𝐰\\boldsymbol\{w\}and𝐯\\boldsymbol\{v\}\.

###### Assumption 3

The local stochastic gradients are unbiased, i\.e\.,

𝖤⁡\[𝒈t\+1k\|𝒘𝒕\]=∇Fk​\(𝒘𝒕\),\\displaystyle\\mathsf\{E\}\[\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\|\\boldsymbol\{w\_\{t\}\}\]=\\nabla F\_\{k\}\(\\boldsymbol\{w\_\{t\}\}\),\(29\)for anyk∈1,…,Mk\\in 1,\\ldots,M\.

###### Assumption 4

The local stochastic gradients are uniformly bounded, i\.e\.,

𝖤​‖𝒈t\+1k‖2≤G2,\\displaystyle\\mathsf\{E\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}\\leq\{G\}^\{2\},\(30\)for anyk∈1,…,Mk\\in 1,\\ldots,M\.

###### Assumption 5

The variances of local stochastic gradients are bounded, i\.e\.,

𝖤​‖𝒈t\+1k−∇Fk​\(𝒘𝒕\)‖2≤σk2,\\displaystyle\\mathsf\{E\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-\\nabla F\_\{k\}\(\\boldsymbol\{w\_\{t\}\}\)\\right\\\|^\{2\}\\leq\\sigma\_\{k\}^\{2\},\(31\)for anyk∈1,…,Mk\\in 1,\\ldots,M\.

We present the following lemmas\[[32](https://arxiv.org/html/2608.13961#bib.bib32)\]\.

###### Lemma 2

Iff⁡\(𝐱\)f\(\\boldsymbol\{x\}\)isLL\-smooth with parameterL1L\_\{1\}andg⁡\(𝐱\)g\(\\boldsymbol\{x\}\)isLL\-smooth with parameterL2L\_\{2\}, then for parametersα≥0\\alpha\\geq 0andβ≥0\\beta\\geq 0, the functionh⁡\(𝐱\)=α​f​\(𝐱\)\+β​g​\(𝐱\)h\(\\boldsymbol\{x\}\)=\\alpha f\(\\boldsymbol\{x\}\)\+\\beta g\(\\boldsymbol\{x\}\)is also L\-smooth with constantα​L1\+β​L2\\alpha L\_\{1\}\+\\beta L\_\{2\}\.

###### Lemma 3

Iff⁡\(𝐱\)f\(\\boldsymbol\{x\}\)isμ\\mu\-strongly convex with constantμ1\{\\mu\}\_\{1\}andg⁡\(𝐱\)g\(\\boldsymbol\{x\}\)isμ\\mu\-strongly convex with constantμ2\{\\mu\}\_\{2\}, then for parametersα≥0\\alpha\\geq 0andβ≥0\\beta\\geq 0, the functionh⁡\(𝐱\)=α​f​\(𝐱\)\+β​g​\(𝐱\)h\(\\boldsymbol\{x\}\)=\\alpha f\(\\boldsymbol\{x\}\)\+\\beta g\(\\boldsymbol\{x\}\)is alsoμ\\mu\-strongly convex with constantα​μ1\+β​μ2\\alpha\{\\mu\}\_\{1\}\+\\beta\{\\mu\}\_\{2\}\.

Based on andAssumption 1,Assumption 2and above lemmas, we conclude that the global loss functionF⁡\(𝒘\)F\(\\boldsymbol\{w\}\)is alsoLL\-smooth with parameterL=1M​∑k=1MLkL=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\{L\}\_\{k\}andμ\\mu\-strongly convex with parameterμ=1M​∑k=1Mμk\\mu=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\{\\mu\}\_\{k\}\.

The convergence analysis of the scheme proposed in the previous section is presented in the following theorem\.

###### Theorem 1

We assume the above assumptions hold\. When the step sizeηt\\eta\_\{t\}satisfies0<ηt≤min⁡\{1,12​L\}0<\\eta\_\{t\}\\leq\\min\\left\\\{1,\\frac\{1\}\{2L\}\\right\\\}for anyt∈\{0,1,2,…,T−1\}t\\in\\\{0,1,2,\\ldots,T\-1\\\}, we have

𝖤⁡\[F⁡\(𝒘T\)−F∗\]≤\\displaystyle\\mathsf\{E\}\\left\[F\\left\(\\boldsymbol\{w\}\_\{T\}\\right\)\-F^\{\*\}\\right\]\\leqL2​\[∏t=1T\(1−μ​ηt\)\]​𝖤​‖𝒘0−𝒘∗‖2\+\\displaystyle\\frac\{L\}\{2\}\\left\[\\prod\_\{t=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{t\}\)\\right\]\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{0\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\+\+L2∑t=1T\[∏j=t\+1T\(1−μηj\)\]ηt2H,\\displaystyle\+\\frac\{L\}\{2\}\\sum\_\{t=1\}^\{T\}\\left\[\\prod\_\{j=t\+1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]\{\\eta\}\_\{t\}^\{2\}H,\(32\)where

H=\\displaystyle H=4​G2\+4​L​Γ\+1M2​∑k=1Mσk2\+d​Δ2​∑i=1n4i−1​Z​\(WN\(i\)\)\+\\displaystyle 4G^\{2\}\+4L\{\\Gamma\}\+\\frac\{1\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\{\\sigma\}\_\{k\}^\{2\}\+d\{\\Delta\}^\{2\}\\sum\_\{i=1\}^\{n\}4^\{i\-1\}Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)\+\+d6​Δ2\+\(M−K\)​G2K⁡\(M−1\)\.\\displaystyle\+\\frac\{d\}\{6\}\{\\Delta\}^\{2\}\+\\frac\{\(M\-K\)G^\{2\}\}\{K\(M\-1\)\}\.\(33\)and

Γ=F∗−1M​∑k=1MFk∗\\displaystyle\\Gamma=F^\{\*\}\-\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}F\_\{k\}^\{\*\}\(34\)wherennis the number of quantization bits,Δ\\Deltais the minimal quantization interval defined in \([10](https://arxiv.org/html/2608.13961#S3.E10)\), andZ⁡\(WN\(i\)\)Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)is the the Bhattacharyya parameter, the upper bound of probability of error of theii\-th bit in the binary sequence\{b0,b1,⋯,bn\}\\left\\\{b\_\{0\},b\_\{1\},\\cdots,b\_\{n\}\\right\\\}\.

We make the following remarks with respect to the above theorem

###### Proof:

We defineΘt=𝖤​‖𝒘t−𝒘∗‖2\{\\Theta\}\_\{t\}=\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}, and the main task is to derive an upper bound ofΘT\{\\Theta\}\_\{T\}\. To simplify the presentation of the proof, we introduce the following variables

𝒖t\+1\\displaystyle\\boldsymbol\{u\}\_\{t\+1\}≜𝒘t−ηt\+1K​∑k∈𝒮t\+1Q⁡\(𝒈t\+1k\)\\displaystyle\\triangleq\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{K\}\\sum\_\{k\\in\\mathcal\{S\}\_\{t\+1\}\}\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\(35\)𝒗t\+1\\displaystyle\\boldsymbol\{v\}\_\{t\+1\}≜𝒘t−ηt\+1K​∑k∈𝒮t\+1𝒈t\+1k,\\displaystyle\\triangleq\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{K\}\\sum\_\{k\\in\\mathcal\{S\}\_\{t\+1\}\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\(36\)𝒗¯t\+1\\displaystyle\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}≜𝒘t−ηt\+1M​∑k=1M𝒈t\+1k,\\displaystyle\\triangleq\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\(37\)and from \([9](https://arxiv.org/html/2608.13961#S2.E9)\), we have

𝒘t\+1=𝒘t−ηt\+1K​∑k∈𝒮t\+1Q~​\(𝒈t\+1k\),\\displaystyle\\boldsymbol\{w\}\_\{t\+1\}=\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{K\}\\sum\_\{k\\in\\mathcal\{S\}\_\{t\+1\}\}\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\),\(38\)where𝒘t\+1\\boldsymbol\{w\}\_\{t\+1\}is the\(t\+1\)\(t\+1\)\-th global parameter after aggregation,𝒖t\+1\\boldsymbol\{u\}\_\{t\+1\}is the “hypothetical” aggregated global parameter before channel transmission,𝒗t\+1\\boldsymbol\{v\}\_\{t\+1\}is the “hypothetical” aggregated global parameter before quantization, and𝒗¯t\+1\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}is the “hypothetical” aggregated global parameter without user scheduling and before quantization\. The variables𝒖t\+1\\boldsymbol\{u\}\_\{t\+1\},𝒗t\+1\\boldsymbol\{v\}\_\{t\+1\}and𝒗¯t\+1\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}are “hypothetical” because they are not actually calculated in the training process\. Based on the above definitions, we can decomposeΘt\+1\\Theta\_\{t\+1\}as follows

Θt\+1=\\displaystyle\{\\Theta\}\_\{t\+1\}=𝖤​‖𝒘t\+1−𝒖t\+1‖2⏟A\+2​𝖤​\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘∗⟩\]\\displaystyle\\underbrace\{\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\}\\right\\\|^\{2\}\}\_\{A\}\+2\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\+\\displaystyle\+𝖤​‖𝒖t\+1−𝒗t\+1‖2⏟B\+2​𝖤​\[⟨𝒖t\+1−𝒗t\+1,𝒗t\+1−𝒘∗⟩\]\\displaystyle\\underbrace\{\\mathsf\{E\}\\left\\\|\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{v\}\_\{t\+1\}\\right\\\|^\{2\}\}\_\{B\}\+2\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{v\}\_\{t\+1\},\\boldsymbol\{v\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\+\\displaystyle\+𝖤​‖𝒗t\+1−𝒗¯t\+1‖2⏟C\+2​𝖤​\[⟨𝒗t\+1−𝒗¯t\+1,𝒗¯t\+1−𝒘∗⟩\]\\displaystyle\\underbrace\{\\mathsf\{E\}\\left\\\|\\boldsymbol\{v\}\_\{t\+1\}\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\\right\\\|^\{2\}\}\_\{C\}\+2\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{v\}\_\{t\+1\}\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\+\\displaystyle\+𝖤​‖𝒗¯t\+1−𝒘∗‖2⏟D\.\\displaystyle\\underbrace\{\\mathsf\{E\}\\left\\\|\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\}\_\{D\}\.\(39\)
To upper boundΘt\+1\\Theta\_\{t\+1\}, we first note that the random variables𝒈t\+1k\\boldsymbol\{g\}\_\{t\+1\}^\{k\},Q⁡\(𝒈t\+1k\)\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)andQ~​\(𝒈t\+1k\)\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)are independent of𝒈t\+1j\\boldsymbol\{g\}\_\{t\+1\}^\{j\},Q⁡\(𝒈t\+1j\)\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\)andQ~​\(𝒈t\+1j\)\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\), for anyk≠jk\\neq j, because the channel noise, the quantization noise and training data are assumed independent in each distributed user\. Therefore, we have

𝖤\\displaystyle\\mathsf\{E\}\[⟨Q⁡\(𝒈t\+1k\)−𝒈t\+1k,Q⁡\(𝒈t\+1j\)−𝒈t\+1j⟩\]=\\displaystyle\[\\langle Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\},Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\\rangle\]=⟨𝖤⁡\[Q⁡\(𝒈t\+1k\)−𝒈t\+1k\],𝖤⁡\[Q⁡\(𝒈t\+1j\)−𝒈t\+1j\]⟩,k≠j\\displaystyle\\langle\\mathsf\{E\}\[Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\],\\mathsf\{E\}\[Q\(\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{j\}\]\\rangle,\\quad k\\neq j\(40\)which leads to the following lemma

###### Lemma 4

For anyt∈\{0,1,2,…,T−1\}t\\in\\\{0,1,2,\\ldots,T\-1\\\},

A≤ηt\+12​d​Δ2​∑i=1n4i−1​Z​\(WN\(i\)\),\\displaystyle A\\leq\{\\eta\}\_\{t\+1\}^\{2\}d\{\\Delta\}^\{2\}\\sum\_\{i=1\}^\{n\}4^\{i\-1\}Z\\left\(W\_\{N\}^\{\(i\)\}\\right\),\(41\)𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘∗⟩\]≤0\.\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\\leq 0\.\(42\)

###### Lemma 5

For anyt∈\{0,1,2,…,T−1\}t\\in\\\{0,1,2,\\ldots,T\-1\\\},

B≤ηt\+12​d6​Δ2\.\\displaystyle B\\leq\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}d\}\{6\}\{\\Delta\}^\{2\}\.\(43\)𝖤⁡\[⟨𝒖t\+1−𝒗t\+1,𝒗t\+1−𝒘∗⟩\]=0\.\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{v\}\_\{t\+1\},\\boldsymbol\{v\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=0\.\(44\)

###### Lemma 6

For anyt∈\{0,1,2,…,T−1\}t\\in\\\{0,1,2,\\ldots,T\-1\\\},

C≤\(M−K\)​ηt\+12​G2K⁡\(M−1\),\\displaystyle C\\leq\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}G^\{2\}\}\{K\(M\-1\)\},\(45\)𝖤⁡\[⟨𝒗t\+1−𝒗¯t\+1,𝒗¯t\+1−𝒘∗⟩\]=0\.\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{v\}\_\{t\+1\}\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=0\.\(46\)

###### Lemma 7

For anyt∈\{0,1,2,…,T−1\}t\\in\\\{0,1,2,\\ldots,T\-1\\\},

D≤\\displaystyle D\\leq\(1−μ​ηt\+1\)​𝖤​‖𝒘t−𝒘∗‖2\+4​ηt\+12​G2\+4​L​ηt\+12​Γ\\displaystyle\(1\-\{\\mu\}\{\\eta\}\_\{t\+1\}\)\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\+4\{\\eta\}\_\{t\+1\}^\{2\}G^\{2\}\+4L\{\\eta\}\_\{t\+1\}^\{2\}\{\\Gamma\}\+ηt\+12M2∑k=1Mσk2\.\\displaystyle\+\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\{\\sigma\}\_\{k\}^\{2\}\.\(47\)

From Lemma[4](https://arxiv.org/html/2608.13961#ThmLem4)to[7](https://arxiv.org/html/2608.13961#ThmLem7), we have

Θt\+1≤\(1−μ​ηt\+1\)​Θt\+ηt\+12​H,\\displaystyle\{\\Theta\}\_\{t\+1\}\\leq\(1\-\{\\mu\}\{\\eta\}\_\{t\+1\}\)\{\\Theta\}\_\{t\}\+\{\\eta\}\_\{t\+1\}^\{2\}H,\(48\)whereHHis defined in \([33](https://arxiv.org/html/2608.13961#S4.E33)\)\. The above recursive upper bound leads to the following bound onΘT\\Theta\_\{T\}which leads to

ΘT≤\[∏i=1T\(1−μ​ηi\)\]​Θ0\+∑i=1T\[∏j=i\+1t\(1−μ​ηj\)\]​ηi2​H,\\displaystyle\{\\Theta\}\_\{T\}\\leq\\left\[\\prod\_\{i=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{i\}\)\\right\]\{\\Theta\}\_\{0\}\+\\sum\_\{i=1\}^\{T\}\\left\[\\prod\_\{j=i\+1\}^\{t\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]\{\\eta\}\_\{i\}^\{2\}H,\(49\)
Due to theAssumption 1that the global loss functionF⁡\(⋅\)F\(\\cdot\)isLL\-smooth, i\.e\., \([27](https://arxiv.org/html/2608.13961#S4.E27)\), and the fact that the gradient at the minimizer∇F​\(𝒘∗\)\\nabla F\(\\boldsymbol\{w^\{\\ast\}\}\)is00, we have

𝖤\\displaystyle\\mathsf\{E\}\[F⁡\(𝒘T\)−F∗\]\\displaystyle\\left\[F\\left\(\\boldsymbol\{w\}\_\{T\}\\right\)\-F^\{\*\}\\right\]≤\(𝒘T−𝒘∗\)T∇Fk\(𝒘∗\)\+L2𝖤‖𝒘T−𝒘∗‖2\\displaystyle\\leq\(\\boldsymbol\{w\}\_\{T\}\-\\boldsymbol\{w\}^\{\\ast\}\)^\{T\}\\nabla F\_\{k\}\(\\boldsymbol\{w\}^\{\\ast\}\)\+\\frac\{L\}\{2\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{T\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=L2​𝖤​‖𝒘T−𝒘∗‖2=L2​ΘT\\displaystyle=\\frac\{L\}\{2\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{T\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=\\frac\{L\}\{2\}\\Theta\_\{T\}\(50\)By combining \([49](https://arxiv.org/html/2608.13961#S4.E49)\) and \([50](https://arxiv.org/html/2608.13961#S4.E50)\), we complete the proof of Theorem 1\. ∎

In the sequel, we will prove Lemma[4](https://arxiv.org/html/2608.13961#ThmLem4)\-[7](https://arxiv.org/html/2608.13961#ThmLem7)\. To facilitate the proof of the following lemmas, we assume that the users are selected uniformly in the scheduling scheme and define

Gt\+1\\displaystyle G\_\{t\+1\}≜\{𝒈t\+1k,k=1,…,M\}\\displaystyle\\triangleq\\\{\\boldsymbol\{g\}\_\{t\+1\}^\{k\},k=1,\\dots,M\\\}\(51\)Gt\+1S\\displaystyle G\_\{t\+1\}^\{S\}≜\{𝒈t\+1k,k∈𝒮t\+1\}\\displaystyle\\triangleq\\\{\\boldsymbol\{g\}\_\{t\+1\}^\{k\},k\\in\\mathcal\{S\}\_\{t\+1\}\\\}\(52\)Gt\+1Q\\displaystyle G\_\{t\+1\}^\{Q\}≜\{Q⁡\(𝒈t\+1k\),k∈𝒮t\+1\}\\displaystyle\\triangleq\\\{\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\),k\\in\\mathcal\{S\}\_\{t\+1\}\\\}\(53\)Gt\+1R\\displaystyle G\_\{t\+1\}^\{R\}≜\{Q~​\(𝒈t\+1k\),k∈𝒮t\+1\}\\displaystyle\\triangleq\\\{\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\),k\\in\\mathcal\{S\}\_\{t\+1\}\\\}\(54\)

### IV\-AProof of Lemma[4](https://arxiv.org/html/2608.13961#ThmLem4)

For brevity here, we simplify the binary sequence\{Bt\+1,j,1k,…,Bt\+1,j,lk,…,Bt\+1,j,nk\}\\\{B\_\{t\+1,j,1\}^\{k\},\\ldots,B\_\{t\+1,j,l\}^\{k\},\\ldots,B\_\{t\+1,j,n\}^\{k\}\\\}as\{B1,…,Bl,…,Bn\}\\\{B\_\{1\},\\ldots,B\_\{l\},\\ldots,B\_\{n\}\\\}denoting the component of the quantized gradientQ⁡\(𝒈t\+1,jk\)\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)in the following derivation\.BlB\_\{l\}equals either 0 or 1\. The bit error rate of the l\-th bit isPlP\_\{l\}and in addition we introduce an indicator random variableIlI\_\{l\}which indicates whether the l\-th bit in the sequence is decoded correctly\.Il=0I\_\{l\}=0means the l\-th bit is the right result andIl=1I\_\{l\}=1means the l\-th bit is flipped\.𝖤⁡\[Il\]=Pl\\mathsf\{E\}\\left\[I\_\{l\}\\right\]=P\_\{l\}\.

We assume that the components of the gradientQ⁡\(𝒈t\+1,jk\)\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)is independent of each other and follow a uniform distribution, so the probability of taking value\{B1,…,Bl,…,Bn\}=\{b1i,…,bli,…,bni\}\\\{B\_\{1\},\\ldots,B\_\{l\},\\ldots,B\_\{n\}\\\}=\\\{b\_\{1\}^\{i\},\\ldots,b\_\{l\}^\{i\},\\ldots,b\_\{n\}^\{i\}\\\}equals12n\\frac\{1\}\{2^\{n\}\}, wherei=1,…,12ni=1,\\ldots,\\frac\{1\}\{2^\{n\}\}\.

Then, we are ready to prove the inequality \([42](https://arxiv.org/html/2608.13961#S4.E42)\)

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘∗⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒘t−𝒘∗⟩\+⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘t⟩\]\.\\displaystyle=\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\+\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}\_\{t\}\\rangle\\right\]\.\(55\)For the first term in \([IV\-A](https://arxiv.org/html/2608.13961#S4.Ex13)\), we have

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒘t−𝒘∗⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤⁡\[⟨∑k∈St\+1\[Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\)\],𝒘t−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\\left\[\\langle\\sum\_\{k\\in S\_\{t\+1\}\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\],\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤𝒘t​\[𝖤⁡\[⟨∑k∈St\+1\[Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\)\],𝒘t−𝒘∗⟩\|𝒘t\]\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{w\}\_\{t\}\}\\left\[\\mathsf\{E\}\\left\[\\langle\\sum\_\{k\\in S\_\{t\+1\}\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\],\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\]\\right\]=𝖤𝒘t​\[⟨𝖤⁡\[∑k∈St\+1\[Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\)\]\|𝒘t\],𝒘t−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{w\}\_\{t\}\}\\left\[\\langle\\mathsf\{E\}\\left\[\\sum\_\{k\\in S\_\{t\+1\}\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\]\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\],\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤𝒘t​\[⟨K​𝖤​\[Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\)\|𝒘t\],𝒘t−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{w\}\_\{t\}\}\\left\[\\langle K\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\],\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\(56\)
From \([26](https://arxiv.org/html/2608.13961#S3.E26)\), we have

𝖤⁡\[Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)\|𝒘t\]\\displaystyle\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\]=𝖤Q~​\(𝒈t\+1,jk\)\|𝒘t​\[𝖤⁡\[Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)\|Q~​\(𝒈t\+1,jk\)\]\|𝒘t\]\\displaystyle=\\mathsf\{E\}\_\{\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\|\\boldsymbol\{w\}\_\{t\}\}\\left\[\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\middle\|\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\]\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\]=∑l=1n2l−1​Δ​𝖤Q~​\(𝒈t\+1,jk\)\|𝒘t​\[𝖤⁡\[B¯l−Bl\|B¯l\]\|𝒘t\]\\displaystyle=\\sum\_\{l=1\}^\{n\}2^\{l\-1\}\{\\Delta\}\\mathsf\{E\}\_\{\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\|\\boldsymbol\{w\}\_\{t\}\}\\left\[\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}\\right\]\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\]\(57\)We note thatB¯l\\bar\{B\}\_\{l\}is the soft channel output through the synthesized channel \([III\-C](https://arxiv.org/html/2608.13961#S3.EGx19)\)\. Due to symmetryB¯l\\bar\{B\}\_\{l\}is uniformly distributed on\{Pl,1−Pl\}\\\{P\_\{l\},1\-P\_\{l\}\\\}

𝖤\\displaystyle\\mathsf\{E\}\[B¯l−Bl\|B¯l\]\\displaystyle\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}\\right\]=12​\(𝖤⁡\[B¯l−Bl\|B¯l=Zl\]\+𝖤⁡\[B¯l−Bl\|B¯l=1−Zl\]\)\\displaystyle=\\frac\{1\}\{2\}\\left\(\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}=Z\_\{l\}\\right\]\+\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}=1\-Z\_\{l\}\\right\]\\right\)\(58\)We assume that due to symmetry,BlB\_\{l\}is uniformly distributed on\{0,1\}\\\{0,1\\\}and

𝖯𝗋⁡\(Bl=0\|B¯l=Zl\)\\displaystyle\\mathsf\{Pr\}\(B\_\{l\}=0\|\\bar\{B\}\_\{l\}=Z\_\{l\}\)=𝖯𝗋⁡\(Bl=1\|B¯l=1−Zl\)=1−Pl\\displaystyle=\\mathsf\{Pr\}\(B\_\{l\}=1\|\\bar\{B\}\_\{l\}=1\-Z\_\{l\}\)=1\-P\_\{l\}\(59\)𝖯𝗋⁡\(Bl=1\|B¯l=Zl\)\\displaystyle\\mathsf\{Pr\}\(B\_\{l\}=1\|\\bar\{B\}\_\{l\}=Z\_\{l\}\)=𝖯𝗋⁡\(Bl=0\|B¯l=1−Zl\)=Pl\\displaystyle=\\mathsf\{Pr\}\(B\_\{l\}=0\|\\bar\{B\}\_\{l\}=1\-Z\_\{l\}\)=P\_\{l\}\(60\)Thus, we have

𝖤⁡\[B¯l−Bl\|B¯l=Pl\]\\displaystyle\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}=P\_\{l\}\\right\]=\(Zl−1\)​Pl\+Zl​\(1−Pl\)\\displaystyle=\(Z\_\{l\}\-1\)P\_\{l\}\+Z\_\{l\}\(1\-P\_\{l\}\)=Zl−Pl\\displaystyle=Z\_\{l\}\-P\_\{l\}\(61\)𝖤⁡\[B¯l−Bl\|B¯l=1−Pl\]\\displaystyle\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}=1\-P\_\{l\}\\right\]=−Zl​\(1−Pl\)\+\(1−Zl\)​Pl\\displaystyle=\-Z\_\{l\}\(1\-P\_\{l\}\)\+\(1\-Z\_\{l\}\)P\_\{l\}=Pl−Zl\\displaystyle=P\_\{l\}\-Z\_\{l\}\(62\)Therefore, from \([58](https://arxiv.org/html/2608.13961#S4.E58)\), \([61](https://arxiv.org/html/2608.13961#S4.E61)\) and \([62](https://arxiv.org/html/2608.13961#S4.E62)\), we have

𝖤⁡\[B¯l−Bl\|B¯l\]\\displaystyle\\mathsf\{E\}\\left\[\\bar\{B\}\_\{l\}\-\{B\}\_\{l\}\\middle\|\\bar\{B\}\_\{l\}\\right\]=0\\displaystyle=0\(63\)which leads to

𝖤⁡\[Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)\|Q~​\(𝒈t\+1,jk\)\]\\displaystyle\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\middle\|\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\]=0\\displaystyle=0\(64\)𝖤⁡\[Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)\|𝒘t\]\\displaystyle\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\middle\|\\boldsymbol\{w\}\_\{t\}\\right\]=0\\displaystyle=0\(65\)and hence

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒘t−𝒘∗⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=0\\displaystyle=0\(66\)
For the second term in equation \([IV\-A](https://arxiv.org/html/2608.13961#S4.Ex13)\), we have

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘t⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}\_\{t\}\\rangle\\right\]=ηt\+12K2​𝖤​\[⟨∑k1∈St\+1\[Q~​\(𝒈t\+1k1\)−Q⁡\(𝒈t\+1k1\)\],∑k2∈St\+1Q⁡\(𝒈t\+1k2\)⟩\]\.\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{\{K\}^\{2\}\}\\mathsf\{E\}\\left\[\\langle\\sum\_\{k\_\{1\}\\in S\_\{t\+1\}\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\\right\],\\sum\_\{k\_\{2\}\\in S\_\{t\+1\}\}\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{2\}\}\)\\rangle\\right\]\.\(67\)
The gradients of different clients are independent of each other, which means that whenk1≠k2k\_\{1\}\\neq k\_\{2\},

𝖤\\displaystyle\\mathsf\{E\}\[⟨Q~​\(𝒈t\+1k1\)−Q⁡\(𝒈t\+1k1\),Q⁡\(𝒈t\+1k2\)⟩\]\\displaystyle\\left\[\\langle\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\),\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{2\}\}\)\\rangle\\right\]=⟨𝖤⁡\[Q~​\(𝒈t\+1k1\)−Q⁡\(𝒈t\+1k1\)\],𝖤⁡\[Q⁡\(𝒈t\+1k2\)\]⟩\\displaystyle=\\langle\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\\right\],\\mathsf\{E\}\\left\[\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{2\}\}\)\\right\]\\rangle=0\\displaystyle=0\(68\)where the last equality is due to \([64](https://arxiv.org/html/2608.13961#S4.E64)\), more specifically

𝖤\\displaystyle\\mathsf\{E\}\[Q~​\(𝒈t\+1k1\)−Q⁡\(𝒈t\+1k1\)\]\\displaystyle\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\\right\]=𝖤Q~​\(𝒈t\+1k1\)​\[𝖤⁡\[Q~​\(𝒈t\+1k1\)−Q⁡\(𝒈t\+1k1\)\|Q~​\(𝒈t\+1k1\)\]\]\\displaystyle=\\mathsf\{E\}\_\{\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\}\\left\[\\mathsf\{E\}\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\|\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\_\{1\}\}\)\\right\]\\right\]=0\.\\displaystyle=0\.\(69\)
So equation \([IV\-A](https://arxiv.org/html/2608.13961#S4.Ex23)\) can be simplified as

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘t⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}\_\{t\}\\rangle\\right\]=ηt\+12K2​𝖤​\[∑k∈St\+1⟨Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\),Q⁡\(𝒈t\+1k\)⟩\]\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{\{K\}^\{2\}\}\\mathsf\{E\}\\left\[\\sum\_\{k\\in S\_\{t\+1\}\}\\langle\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\),\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\rangle\\right\]=ηt\+12K​𝖤​\[⟨Q~​\(𝒈t\+1k\)−Q⁡\(𝒈t\+1k\),Q⁡\(𝒈t\+1k\)⟩\]\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{\{K\}\}\\mathsf\{E\}\\left\[\\langle\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\),\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\rangle\\right\]=ηt\+12​dK​𝖤​\[\[Q~​\(𝒈t\+1,jk\)−Q⁡\(𝒈t\+1,jk\)\]​Q​\(𝒈t\+1,jk\)\]\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}d\}\{\{K\}\}\\mathsf\{E\}\\left\[\\left\[\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\]\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\]=ηt\+12​dK​𝖤​\[\(∑l=1n\(B¯l−Bl\)​2l−1​Δ\)​\(∑l=1nBl​2l−1​Δ\)\]\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}d\}\{\{K\}\}\\mathsf\{E\}\\left\[\\left\(\\sum\_\{l=1\}^\{n\}\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)2^\{l\-1\}\{\\Delta\}\\right\)\\left\(\\sum\_\{l=1\}^\{n\}B\_\{l\}2^\{l\-1\}\{\\Delta\}\\right\)\\right\]=ηt\+12​d​Δ2K​𝖤​\[\(∑l=1n4l−1​\(B¯l−Bl\)​Bl\)\]\.\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}d\{\\Delta^\{2\}\}\}\{\{K\}\}\\mathsf\{E\}\\left\[\\left\(\\sum\_\{l=1\}^\{n\}4^\{l\-1\}\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\\right\)\\right\]\.\(70\)where the last equality is because of the following derivation\. We assume that the channel inputsBlB\_\{l\}, forl=1,…,nl=1,\\dots,n, are independent of each other\. Therefore for1≤i≠j≤n1\\leq i\\neq j\\leq n, we have

𝖤\\displaystyle\\mathsf\{E\}\(\(B¯i−Bi\)​Bj\)\\displaystyle\(\(\\bar\{B\}\_\{i\}\-B\_\{i\}\)B\_\{j\}\)=𝖤⁡\(B¯i−Bi\)​𝖤​Bj\\displaystyle=\\mathsf\{E\}\(\\bar\{B\}\_\{i\}\-B\_\{i\}\)\\mathsf\{E\}B\_\{j\}=𝖤B¯i​\[𝖤⁡\(B¯i−Bi\|B¯i\)\]​𝖤​Bj\\displaystyle=\\mathsf\{E\}\_\{\\bar\{B\}\_\{i\}\}\[\\mathsf\{E\}\(\\bar\{B\}\_\{i\}\-B\_\{i\}\|\\bar\{B\}\_\{i\}\)\]\\mathsf\{E\}B\_\{j\}=0\\displaystyle=0\(71\)where the last equality is due to \([63](https://arxiv.org/html/2608.13961#S4.E63)\)\. Furthermore, for the term𝖤⁡\(\(B¯l−Bl\)​Bl\)\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\), we have

𝖤\\displaystyle\\mathsf\{E\}\(\(B¯l−Bl\)​Bl\)=𝖤Bl​\[𝖤⁡\(\(B¯l−Bl\)​Bl\|Bl\)\]\\displaystyle\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\)=\\mathsf\{E\}\_\{B\_\{l\}\}\[\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\|B\_\{l\}\)\]=12​\[𝖤⁡\(\(B¯l−Bl\)​Bl\|Bl=0\)\+𝖤⁡\(\(B¯l−Bl\)​Bl\|Bl=1\)\]\\displaystyle=\\frac\{1\}\{2\}\[\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\|B\_\{l\}=0\)\+\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)B\_\{l\}\|B\_\{l\}=1\)\]=12\[0\+\(Zl−1\)Pl−Zl\(1−Pl\)\)\]\\displaystyle=\\frac\{1\}\{2\}\\left\[0\+\(Z\_\{l\}\-1\)P\_\{l\}\-Z\_\{l\}\(1\-P\_\{l\}\)\)\\right\]=Pl​\(Zl−1\)\+Zl​\(Pl−1\)\\displaystyle=P\_\{l\}\(Z\_\{l\}\-1\)\+Z\_\{l\}\(P\_\{l\}\-1\)≤0\\displaystyle\\leq 0\(72\)Therefore, we have

𝖤⁡\[⟨𝒘t\+1−𝒖t\+1,𝒖t\+1−𝒘t⟩\]≤0\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\},\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{w\}\_\{t\}\\rangle\\right\]\\leq 0\(73\)
By combining \([66](https://arxiv.org/html/2608.13961#S4.E66)\) and \([73](https://arxiv.org/html/2608.13961#S4.E73)\), we prove \([42](https://arxiv.org/html/2608.13961#S4.E42)\)\.

For the termAA, we have

A\\displaystyle A=𝖤​‖𝒘t\+1−𝒖t\+1‖2\\displaystyle=\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\+1\}\-\\boldsymbol\{u\}\_\{t\+1\}\\right\\\|^\{2\}=ηt\+12K2​𝖤​‖∑k∈St\+1\[Q⁡\(𝒈t\+1k\)−Q~​\(𝒈t\+1k\)\]‖2\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{\{K\}^\{2\}\}\\mathsf\{E\}\\left\\\|\\sum\_\{k\\in S\_\{t\+1\}\}\\left\[\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\]\\right\\\|^\{2\}=ηt\+12​𝖤​‖∑k∈St\+11K​\[Q⁡\(𝒈t\+1k\)−Q~​\(𝒈t\+1k\)\]‖2\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\\\|\\sum\_\{k\\in S\_\{t\+1\}\}\\frac\{1\}\{K\}\\left\[\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\]\\right\\\|^\{2\}≤\(a\)ηt\+12​𝖤​\[∑k∈St\+11K​‖Q⁡\(𝒈t\+1k\)−Q~​\(𝒈t\+1k\)‖2\]\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\[\\sum\_\{k\\in S\_\{t\+1\}\}\\frac\{1\}\{K\}\\left\\\|\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\\\|^\{2\}\\right\]=ηt\+12​𝖤​\[‖Q⁡\(𝒈t\+1k\)−Q~​\(𝒈t\+1k\)‖2\]\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\[\\left\\\|\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\\right\\\|^\{2\}\\right\]=ηt\+12​d​𝖤​\[\(Q⁡\(𝒈t\+1,jk\)−Q~​\(𝒈t\+1,jk\)\)2\]\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}d\\mathsf\{E\}\\left\[\\left\(\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\)^\{2\}\\right\]=ηt\+12​d​𝖤​\[\(∑l=1n\(B¯l−Bl\)​2l−1​Δ\)2\]\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}d\\mathsf\{E\}\\left\[\\left\(\\sum\_\{l=1\}^\{n\}\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)2^\{l\-1\}\{\\Delta\}\\right\)^\{2\}\\right\]=ηt\+12​d​Δ2​∑l=1n𝖤⁡\[\(B¯l−Bl\)2\]​4l−1,\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}d\{\\Delta\}^\{2\}\\sum\_\{l=1\}^\{n\}\\mathsf\{E\}\\left\[\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\\right\]4^\{l\-1\},\(74\)where inequalities in \(a\) are due to Jensen’s inequality, and the last equality is due to \([71](https://arxiv.org/html/2608.13961#S4.E71)\)\.

For the term𝖤⁡\[\(B¯l−Bl\)2\]\\mathsf\{E\}\\left\[\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\\right\], we have

𝖤\\displaystyle\\mathsf\{E\}\[\(B¯l−Bl\)2\]\\displaystyle\\left\[\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\\right\]=𝖤Bl​\[𝖤⁡\(\(B¯l−Bl\)2\|Bl\)\]\\displaystyle=\\mathsf\{E\}\_\{B\_\{l\}\}\[\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\|B\_\{l\}\)\]=12​\[𝖤⁡\(\(B¯l−Bl\)2\|Bl=0\)\+𝖤⁡\(\(B¯l−Bl\)2\|Bl=1\)\]\\displaystyle=\\frac\{1\}\{2\}\[\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\|B\_\{l\}=0\)\+\\mathsf\{E\}\(\(\\bar\{B\}\_\{l\}\-B\_\{l\}\)^\{2\}\|B\_\{l\}=1\)\]=12​\[Zl2​\(1−Pl\)\+\(1−Zl\)2​Pl\+\(Zl−1\)2​Pl\+Zl2​\(1−Pl\)\]\\displaystyle=\\frac\{1\}\{2\}\[Z\_\{l\}^\{2\}\(1\-P\_\{l\}\)\+\(1\-Z\_\{l\}\)^\{2\}P\_\{l\}\+\(Z\_\{l\}\-1\)^\{2\}P\_\{l\}\+Z\_\{l\}^\{2\}\(1\-P\_\{l\}\)\]=Pl\+Zl2−2​Zl​Pl\\displaystyle=P\_\{l\}\+Z\_\{l\}^\{2\}\-2Z\_\{l\}P\_\{l\}=Pl​\(1−Pl\)\+\(Zl−Pl\)2\\displaystyle=P\_\{l\}\(1\-P\_\{l\}\)\+\(Z\_\{l\}\-P\_\{l\}\)^\{2\}≤Pl\+\(Zl−Pl\)\\displaystyle\\leq P\_\{l\}\+\(Z\_\{l\}\-P\_\{l\}\)=Zl\.\\displaystyle=Z\_\{l\}\.\(75\)
Since we takeZl=Z⁡\(WN\(l\)\)Z\_\{l\}=Z\\left\(W\_\{N\}^\{\(l\)\}\\right\), then we have

A≤ηt\+12​d​Δ2​∑i=1n4i−1​Z​\(WN\(i\)\)\.\\displaystyle A\\leq\{\\eta\}\_\{t\+1\}^\{2\}d\{\\Delta\}^\{2\}\\sum\_\{i=1\}^\{n\}4^\{i\-1\}Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)\.\(76\)

### IV\-BProof of Lemma[5](https://arxiv.org/html/2608.13961#ThmLem5)

According to Lemma 1, we have

𝖤⁡\[𝒖t\+1\|Gt\+1S\]=𝒗t\+1,\\displaystyle\\mathsf\{E\}\\left\[\\boldsymbol\{u\}\_\{t\+1\}\|G\_\{t\+1\}^\{S\}\\right\]=\\boldsymbol\{v\}\_\{t\+1\},\(77\)and according to the similar argument as in \([IV\-C](https://arxiv.org/html/2608.13961#S4.Ex64)\), we have

𝖤\\displaystyle\\mathsf\{E\}\[⟨𝒖t\+1−𝒗t\+1,𝒗t\+1−𝒘∗⟩\]\\displaystyle\\left\[\\langle\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{v\}\_\{t\+1\},\\boldsymbol\{v\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤Gt\+1S​\[𝖤⁡\[⟨𝒖t\+1−𝒗t\+1,𝒗t\+1−𝒘∗⟩\|Gt\+1S\]\]\\displaystyle=\\mathsf\{E\}\_\{G\_\{t\+1\}^\{S\}\}\\left\[\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{u\}\_\{t\+1\}\-\\boldsymbol\{v\}\_\{t\+1\},\\boldsymbol\{v\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\middle\|G\_\{t\+1\}^\{S\}\\right\]\\right\]=𝖤Gt\+1S​\[⟨𝖤⁡\[𝒖t\+1\|Gt\+1S\]−𝒗t\+1,𝒗t\+1−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\_\{G\_\{t\+1\}^\{S\}\}\\left\[\\langle\\mathsf\{E\}\\left\[\\boldsymbol\{u\}\_\{t\+1\}\\middle\|G\_\{t\+1\}^\{S\}\\right\]\-\\boldsymbol\{v\}\_\{t\+1\},\\boldsymbol\{v\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=0\.\\displaystyle=0\.\(78\)which proves \([44](https://arxiv.org/html/2608.13961#S4.E44)\)\.

Then we have

B\\displaystyle B=ηt\+12K2​𝖤​‖∑k∈St\+1\[Q⁡\(𝒈t\+1k\)−𝒈t\+1k\]‖2\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{\{K\}^\{2\}\}\\mathsf\{E\}\\left\\\|\\sum\_\{k\\in S\_\{t\+1\}\}\\left\[\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\]\\right\\\|^\{2\}=ηt\+12​𝖤​‖∑k∈St\+11K​\[Q⁡\(𝒈t\+1k\)−𝒈t\+1k\]‖2\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\\\|\\sum\_\{k\\in S\_\{t\+1\}\}\\frac\{1\}\{K\}\\left\[\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\]\\right\\\|^\{2\}≤\(a\)ηt\+12​𝖤​\[∑k∈St\+11K​‖Q⁡\(𝒈t\+1k\)−𝒈t\+1k‖2\]\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\[\\sum\_\{k\\in S\_\{t\+1\}\}\\frac\{1\}\{K\}\\left\\\|\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}\\right\]=ηt\+12​𝖤​‖Q⁡\(𝒈t\+1k\)−𝒈t\+1k‖2\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\\\|\{Q\}\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}=ηt\+12​d​𝖤​\[\(Q⁡\(𝒈t\+1,jk\)−𝒈t\+1,jk\)2\],\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}d\\mathsf\{E\}\\left\[\\left\(\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\right\)^\{2\}\\right\],\(79\)where \(a\) is because Jensen’s inequality\.

For the unbiased stochastic quantization method, assuming that𝒈t\+1,jk∈\[si,si\+1\]\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\in\[s\_\{i\},s\_\{i\+1\}\]and we have

Q⁡\(𝒈t\+1,jk\)=\{si,w\.p\.​si\+1−𝒈t\+1,jksi\+1−si=psi\+1,w\.p\.​𝒈t\+1,jk−sisi\+1−si=1−p\\displaystyle Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)=\\left\\\{\\begin\{array\}\[\]\{ll\}s\_\{i\},&\\text\{ w\.p\. \}\\frac\{s\_\{i\+1\}\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\}\{s\_\{i\+1\}\-s\_\{i\}\}=p\\\\ s\_\{i\+1\},&\\text\{ w\.p\. \}\\frac\{\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\-s\_\{i\}\}\{s\_\{i\+1\}\-s\_\{i\}\}=1\-p\\end\{array\}\\right\.wherep∈\[0,1\]p\\in\[0,1\]and thussi\+1−𝒈t\+1,jk=p​Δs\_\{i\+1\}\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}=p\{\\Delta\}and𝒈t\+1,jk−si=\(1−p\)​Δ\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\-s\_\{i\}=\(1\-p\)\{\\Delta\}\. So the term𝖤⁡\[\(Q⁡\(𝒈t\+1,jk\)−𝒈t\+1,jk\)2\]\\mathsf\{E\}\\left\[\\left\(\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\right\)^\{2\}\\right\]can be written as

𝖤\\displaystyle\\mathsf\{E\}\[\(Q⁡\(𝒈t\+1,jk\)−𝒈t\+1,jk\)2\]\\displaystyle\\left\[\\left\(\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\right\)^\{2\}\\right\]=𝖤𝒈t\+1,jk​\[𝖤⁡\[\(Q⁡\(𝒈t\+1,jk\)−𝒈t\+1,jk\)2∣𝒈t\+1,jk\]\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\}\\Bigg\[\\mathsf\{E\}\\left\[\\left\(\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\right\)^\{2\}\\mid\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\\right\]\\Bigg\]=∫01\[p​\(1−p\)2​Δ2\+\(1−p\)​p2​Δ2\]​𝑑p\\displaystyle=\\int\_\{0\}^\{1\}\\Big\[p\(1\-p\)^\{2\}\{\\Delta\}^\{2\}\+\(1\-p\)p^\{2\}\{\\Delta\}^\{2\}\\Big\]\\mathrm\{d\}p=16​Δ2\.\\displaystyle=\\frac\{1\}\{6\}\{\\Delta\}^\{2\}\.\(82\)which completes the proof of Lemma[5](https://arxiv.org/html/2608.13961#ThmLem5)\.

### IV\-CProof of Lemma[6](https://arxiv.org/html/2608.13961#ThmLem6)

We have

𝖤⁡\[1K​∑k∈𝒮t\+1𝒈t\+1k\|Gt\+1\]\\displaystyle\\mathsf\{E\}\\left\[\\frac\{1\}\{K\}\\sum\_\{k\\in\\mathcal\{S\}\_\{t\+1\}\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\middle\|G\_\{t\+1\}\\right\]=1M​∑k=1M𝒈t\+1k,\\displaystyle=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\(83\)which implies

𝖤⁡\[𝒗t\+1\|Gt\+1\]=𝒗¯t\+1,\\displaystyle\\mathsf\{E\}\\left\[\\boldsymbol\{v\}\_\{t\+1\}\|G\_\{t\+1\}\\right\]=\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\(84\)and therefore

𝖤⁡\[⟨𝒗t\+1−𝒗¯t\+1,𝒗¯t\+1−𝒘∗⟩\]\\displaystyle\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{v\}\_\{t\+1\}\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=𝖤Gt\+1​\[𝖤⁡\[⟨𝒗t\+1−𝒗¯t\+1,𝒗¯t\+1−𝒘∗⟩\|Gt\+1\]\]\\displaystyle=\\mathsf\{E\}\_\{G\_\{t\+1\}\}\\left\[\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{v\}\_\{t\+1\}\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\middle\|G\_\{t\+1\}\\right\]\\right\]=𝖤Gt\+1​\[⟨𝖤⁡\[𝒗t\+1\|Gt\+1\]−𝒗¯t\+1,𝒗¯t\+1−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\_\{G\_\{t\+1\}\}\\left\[\\langle\\mathsf\{E\}\\left\[\\boldsymbol\{v\}\_\{t\+1\}\|G\_\{t\+1\}\\right\]\-\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\},\\overline\{\\boldsymbol\{v\}\}\_\{t\+1\}\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]=0\.\\displaystyle=0\.\(85\)which proves \([46](https://arxiv.org/html/2608.13961#S4.E46)\)\.

We define𝒈¯t\+1≜1M​∑k=1M𝒈t\+1k\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\triangleq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\. We also define the indicator functionI\{k∈St\+1\}I\\\{k\\in S\_\{t\+1\}\\\}asI\{k∈St\+1\}=1I\\\{k\\in S\_\{t\+1\}\\\}=1whenk∈St\+1k\\in S\_\{t\+1\}andI\{k∈St\+1\}=0I\\\{k\\in S\_\{t\+1\}\\\}=0whenk∉St\+1k\\notin S\_\{t\+1\}\. We note that∑k=1MI\{k∈St\+1\}=K\\sum\_\{k=1\}^\{M\}I\\\{k\\in S\_\{t\+1\}\\\}=K\. The proof of the lemma is as follows

C=\\displaystyle C=ηt\+12​𝖤​‖1K​∑k∈St\+1𝒈t\+1k−1M​∑k=1M𝒈t\+1k‖2\\displaystyle\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\\\|\\frac\{1\}\{K\}\\sum\_\{k\\in S\_\{t\+1\}\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}=\\displaystyle=ηt\+12K2​𝖤​‖∑k∈St\+1𝒈t\+1k−K​𝒈¯t\+1‖2\\displaystyle\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{K^\{2\}\}\\mathsf\{E\}\\left\\\|\\sum\_\{k\\in S\_\{t\+1\}\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-K\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}=\\displaystyle=ηt\+12K2𝖤‖∑k=1M\[I\{k∈St\+1\}\(𝒈t\+1k−𝒈¯t\+1\)\]‖2\\displaystyle\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{K^\{2\}\}\\mathsf\{E\}\\left\\\|\\sum\_\{k=1\}^\{M\}\\Big\[I\\\{k\\in S\_\{t\+1\}\\\}\\left\(\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\)\\Big\]\\right\\\|^\{2\}=\(a\)\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{=\}\}\(M−K\)​ηt\+12M​K​\(M−1\)​𝖤​\[∑k=1M‖𝒈t\+1k−𝒈¯t\+1‖2\]\\displaystyle\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}\}\{MK\(M\-1\)\}\\mathsf\{E\}\\left\[\\sum\_\{k=1\}^\{M\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}\\right\]=\\displaystyle=\(M−K\)​ηt\+12M​K​\(M−1\)\[∑k=1M𝖤‖𝒈t\+1k‖2−2∑k=1M𝖤\[⟨𝒈t\+1k,𝒈¯t\+1⟩\]\\displaystyle\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}\}\{MK\(M\-1\)\}\\Bigg\[\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}\-2\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\rangle\\right\]\+∑k=1M𝖤‖𝒈¯t\+1‖2\]\\displaystyle\+\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}\\Bigg\]=\(b\)\\displaystyle\\stackrel\{\{\\scriptstyle\(b\)\}\}\{\{=\}\}\(M−K\)​ηt\+12M​K​\(M−1\)​\[∑k=1M𝖤​‖𝒈t\+1k‖2−\(2​M−1\)​𝖤​‖𝒈¯t\+1‖2\]\\displaystyle\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}\}\{MK\(M\-1\)\}\\Bigg\[\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}\-\(2M\-1\)\\mathsf\{E\}\\left\\\|\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}\\Bigg\]≤\\displaystyle\\leq\(M−K\)​ηt\+12M​K​\(M−1\)​∑k=1M𝖤​‖𝒈t\+1k‖2\\displaystyle\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}\}\{MK\(M\-1\)\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}=\(c\)\\displaystyle\\stackrel\{\{\\scriptstyle\(c\)\}\}\{\{=\}\}\(M−K\)​ηt\+12​G2K⁡\(M−1\)\.\\displaystyle\\frac\{\(M\-K\)\{\\eta\}\_\{t\+1\}^\{2\}G^\{2\}\}\{K\(M\-1\)\}\.\(86\)where \(a\) is from Lemma 4 in\[[26](https://arxiv.org/html/2608.13961#bib.bib26)\]; \(b\) is because of∑k=1M𝖤⁡\[⟨𝒈t\+1k,𝒈¯t\+1⟩\]=M​𝖤​‖𝒈¯t\+1‖2\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{g\}\_\{t\+1\}^\{k\},\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\rangle\\right\]=M\\mathsf\{E\}\\left\\\|\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}; \(c\) is due toAssumption 4\.

### IV\-DProof of Lemma[7](https://arxiv.org/html/2608.13961#ThmLem7)

We define∇F​\(𝒘t\)≜1M​∑k=1M∇Fk​\(𝒘t\)\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\\triangleq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\), and the lemma is proved as follows

D=𝖤​‖𝒘t−ηt\+1M​∑k=1M𝒈t\+1k−𝒘∗‖2\\displaystyle D=\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\frac\{\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=𝖤‖𝒘t−ηt\+1𝒈¯t\+1\+ηt\+1∇F\(𝒘t\)−ηt\+1∇F\(𝒘t\)−𝒘∗‖2\\displaystyle=\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\+\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=𝖤‖𝒘t−ηt\+1∇F\(𝒘t\)−𝒘∗‖2⏟D1\+ηt\+12​𝖤​‖∇F​\(𝒘t\)−𝒈¯t\+1‖2⏟D2\\displaystyle=\\underbrace\{\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\}\_\{D\_\{1\}\}\+\\underbrace\{\{\\eta\}\_\{t\+1\}^\{2\}\\mathsf\{E\}\\left\\\|\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\\\|^\{2\}\}\_\{D\_\{2\}\}\+2​𝖤\[⟨ηt\+1\[∇F\(𝒘t\)−𝒈¯t\+1\],𝒘t−ηt\+1∇F\(𝒘t\)−𝒘∗⟩\]⏟Q\.\\displaystyle\+2\\underbrace\{\\mathsf\{E\}\\left\[\\langle\{\\eta\}\_\{t\+1\}\\left\[\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\],\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{w\}^\{\*\}\\rangle\\right\]\}\_\{Q\}\.\(87\)
For the last term in the above equation, we have

Q\\displaystyle Q=𝖤𝒘t\[𝖤\[⟨ηt\+1\[∇F\(𝒘t\)−𝒈¯t\+1\],𝒘t−ηt\+1∇F\(𝒘t\)−𝒘∗⟩∣𝒘t\]\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{w\}\_\{t\}\}\\Bigg\[\\mathsf\{E\}\\Big\[\\langle\{\\eta\}\_\{t\+1\}\\left\[\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\right\],\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{w\}^\{\*\}\\rangle\\mid\\boldsymbol\{w\}\_\{t\}\\Big\]\\Bigg\]=𝖤𝒘t\[⟨ηt\+1𝖤\[∇F\(𝒘t\)−𝒈¯t\+1∣𝒘t\],𝒘t−ηt\+1∇F\(𝒘t\)−𝒘∗⟩\]\\displaystyle=\\mathsf\{E\}\_\{\\boldsymbol\{w\}\_\{t\}\}\\Bigg\[\\langle\{\\eta\}\_\{t\+1\}\\mathsf\{E\}\\left\[\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\overline\{\\boldsymbol\{g\}\}\_\{t\+1\}\\mid\\boldsymbol\{w\}\_\{t\}\\right\],\\boldsymbol\{w\}\_\{t\}\-\{\\eta\}\_\{t\+1\}\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{w\}^\{\*\}\\rangle\\Bigg\]=0\.\\displaystyle=0\.\(88\)where the last equality is dueAssumption 4\.

For the termD2D\_\{2\}, we have

D2\\displaystyle D\_\{2\}=ηt\+12M2​∑k=1M𝖤​‖∇Fk​\(𝒘t\)−𝒈t\+1k‖2\\displaystyle=\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-\\boldsymbol\{g\}\_\{t\+1\}^\{k\}\\right\\\|^\{2\}≤ηt\+12M2​∑k=1Mσk2\.\\displaystyle\\leq\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\\sigma\_\{k\}^\{2\}\.\(89\)where the last inequality is due toAssumption 5\.

It remains to upper bound the termD1D\_\{1\}as follows

D1\\displaystyle D\_\{1\}=𝖤​‖𝒘t−𝒘∗‖2​−2​ηt\+1​𝖤​\[⟨𝒘t−𝒘∗,∇F​\(𝒘t\)⟩\]⏟P1\\displaystyle=\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\\underbrace\{\-2\{\\eta\}\_\{t\+1\}\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\},\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\\rangle\\right\]\}\_\{P\_\{1\}\}\+ηt\+12​‖∇F​\(𝒘t\)‖2⏟P2\.\\displaystyle\\quad\+\\underbrace\{\{\\eta\}\_\{t\+1\}^\{2\}\\left\\\|\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}\}\_\{P\_\{2\}\}\.\(90\)For the termP2P\_\{2\}, we have

P2\\displaystyle P\_\{2\}=ηt\+12​‖∇F​\(𝒘t\)‖2\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\left\\\|\\nabla F\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}=ηt\+12​‖1M​∑k=1M∇Fk​\(𝒘t\)‖2\\displaystyle=\{\\eta\}\_\{t\+1\}^\{2\}\\left\\\|\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}≤\(a\)ηt\+12M​∑k=1M‖∇Fk​\(𝒘t\)‖2\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}≤\(b\)2​L​ηt\+12M​∑k=1M\(Fk​\(𝒘t\)−Fk∗\)\.\\displaystyle\\stackrel\{\{\\scriptstyle\(b\)\}\}\{\{\\leq\}\}\\frac\{2L\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\left\(F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}^\{\*\}\\right\)\.\(91\)where \(a\) follows Jensen’s inequality and \(b\) uses theLL\-smooth property ofFk​\(⋅\)F\_\{k\}\(\\cdot\)by substituting the following inequality

‖∇Fk​\(𝒘t\)‖2≤2​L​\(Fk​\(𝒘t\)−Fk∗\),\\displaystyle\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}\\leq 2L\\left\(F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}^\{\*\}\\right\),\(92\)whereFk∗F\_\{k\}^\{\*\}represents the optimal value of the local loss functionFk​\(⋅\)F\_\{k\}\(\\cdot\)\. Next,

P1\\displaystyle P\_\{1\}=−2​ηt\+1M∑k=1M𝖤\[⟨𝒘t−𝒘∗,∇Fk\(𝒘t\)⟩\]\\displaystyle=\-\\frac\{2\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[\\langle\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\},\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\rangle\\right\]=−2​ηt\+1M∑k=1M𝖤\[⟨𝒘t−𝒘tk,∇Fk\(𝒘t\)⟩\+\\displaystyle=\-\\frac\{2\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\big\[\\langle\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\},\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\rangle\+⟨𝒘tk−𝒘∗,∇Fk\(𝒘t\)⟩\]\.\\displaystyle\\quad\\langle\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\},\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\rangle\\big\]\.\(93\)
And

−2​⟨𝒘t−𝒘tk,∇Fk​\(𝒘t\)⟩\\displaystyle\-2\\langle\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\},\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\rangle≤\(a\)2​‖𝒘t−𝒘tk‖​‖∇Fk​\(𝒘t\)‖\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}2\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|≤\(b\)1ηt\+1​‖𝒘t−𝒘tk‖2\+ηt\+1​‖∇Fk​\(𝒘t\)‖2,\\displaystyle\\stackrel\{\{\\scriptstyle\(b\)\}\}\{\{\\leq\}\}\\frac\{1\}\{\{\\eta\}\_\{t\+1\}\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|^\{2\}\+\{\\eta\}\_\{t\+1\}\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\},\(94\)where \(a\) follows Cauchy inequality and \(b\) follows AM\-GM inequality\. Then

−⟨𝒘tk−𝒘∗,∇Fk​\(𝒘t\)⟩≤\(a\)\\displaystyle\-\\langle\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\},\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\rangle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}−\[Fk​\(𝒘t\)−Fk​\(𝒘∗\)\]\\displaystyle\-\\left\[F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)\\right\]−μ2​‖𝒘tk−𝒘∗‖2,\\displaystyle\-\\frac\{\\mu\}\{2\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\},\(95\)where \(a\) uses theμ\\mu\-strongly convex property ofFk​\(⋅\)F\_\{k\}\(\\cdot\)\. Substituting equation \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex88)\) and \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex89)\) into \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex86)\), we get that

P1\\displaystyle P\_\{1\}≤1M​∑k=1M𝖤​‖𝒘t−𝒘tk‖2−2​ηt\+1M​∑k=1M𝖤⁡\[Fk​\(𝒘t\)−Fk​\(𝒘∗\)\]\\displaystyle\\leq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|^\{2\}\-\\frac\{2\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)\\right\]\+ηt\+12M∑k=1M‖∇Fk\(𝒘t\)‖2−μ​ηt\+1M∑k=1M𝖤‖𝒘tk−𝒘∗‖2\.\\displaystyle\\quad\+\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\left\\\|\\nabla F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\\right\\\|^\{2\}\-\\frac\{\\mu\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\.\(96\)
Since

1M​∑k=1M𝖤​‖𝒘t−𝒘∗‖2=1M​∑k=1M𝖤​‖1M​∑k=1M\(𝒘tk−𝒘∗\)‖2,\\displaystyle\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\left\(\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\)\\right\\\|^\{2\},\(97\)and following Jensen’s inequality,

1M​∑k=1M𝖤​‖𝒘t−𝒘∗‖2\\displaystyle\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}≤1M​∑k=1M∑k=1M1M​𝖤​‖𝒘tk−𝒘∗‖2\\displaystyle\\leq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\sum\_\{k=1\}^\{M\}\\frac\{1\}\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}=1M​∑k=1M𝖤​‖𝒘tk−𝒘∗‖2,\\displaystyle=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\},\(98\)we can get that

−μ​ηt\+1M∑k=1M𝖤‖𝒘tk−𝒘∗‖2≤−μ​ηt\+1M∑k=1M𝖤‖𝒘t−𝒘∗‖2\.\\displaystyle\-\\frac\{\\mu\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\\leq\-\\frac\{\\mu\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\.\(99\)
Substituting equation \([92](https://arxiv.org/html/2608.13961#S4.E92)\) and equation \([99](https://arxiv.org/html/2608.13961#S4.E99)\) into equation \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex90)\), we obtain that

P1\+P2\\displaystyle P\_\{1\}\+P\_\{2\}≤1M​∑k=1M𝖤​‖𝒘t−𝒘tk‖2−μ​ηt\+1M​∑k=1M𝖤​‖𝒘t−𝒘∗‖2\+\\displaystyle\\leq\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|^\{2\}\-\\frac\{\\mu\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\+4​L​ηt\+12M​∑k=1M𝖤⁡\[Fk​\(𝒘t\)−Fk∗\]−2​ηt\+1M​∑k=1M𝖤⁡\[Fk​\(𝒘t\)−Fk​\(𝒘∗\)\]⏟R\.\\displaystyle\\underbrace\{\\frac\{4L\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}^\{\*\}\\right\]\-\\frac\{2\{\\eta\}\_\{t\+1\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)\\right\]\}\_\{R\}\.\(100\)
And

R\\displaystyle R=\(4​L​ηt\+12M−2​ηt\+1M\)​∑k=1M𝖤⁡\[Fk​\(𝒘t\)−Fk​\(𝒘∗\)\]\+\\displaystyle=\\left\(\\frac\{4L\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\-\\frac\{2\{\\eta\}\_\{t\+1\}\}\{M\}\\right\)\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[F\_\{k\}\(\\boldsymbol\{w\}\_\{t\}\)\-F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)\\right\]\+4​L​ηt\+12M​∑k=1M𝖤⁡\[Fk​\(𝒘∗\)−Fk∗\]\\displaystyle\\quad\\ \\frac\{4L\{\\eta\}\_\{t\+1\}^\{2\}\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\[F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)\-F\_\{k\}^\{\*\}\\right\]=2​ηt\+1​\(2​L​ηt\+1−1\)​\[F⁡\(𝒘t\)−F⁡\(𝒘∗\)\]\+4​L​ηt\+12​Γ,\\displaystyle=2\{\\eta\}\_\{t\+1\}\\left\(2L\{\\eta\}\_\{t\+1\}\-1\\right\)\\left\[F\(\\boldsymbol\{w\}\_\{t\}\)\-F\(\\boldsymbol\{w\}^\{\*\}\)\\right\]\+4L\{\\eta\}\_\{t\+1\}^\{2\}\{\\Gamma\},\(101\)whereF⁡\(𝒘∗\)=1M​∑k=1MFk​\(𝒘∗\)F\(\\boldsymbol\{w\}^\{\*\}\)=\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}F\_\{k\}\(\\boldsymbol\{w\}^\{\*\}\)andΓ=F⁡\(𝒘∗\)−1M​∑k=1MFk∗\\Gamma=F\(\\boldsymbol\{w\}^\{\*\}\)\-\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}F\_\{k\}^\{\*\}\. SinceF⁡\(𝒘t\)−F⁡\(𝒘∗\)≥0F\(\\boldsymbol\{w\}\_\{t\}\)\-F\(\\boldsymbol\{w\}^\{\*\}\)\\geq 0and we letηt\+1≤12​L\{\\eta\}\_\{t\+1\}\\leq\\frac\{1\}\{2L\}, we can obtain that

R≤4​L​ηt\+12​Γ\.\\displaystyle R\\leq 4L\{\\eta\}\_\{t\+1\}^\{2\}\{\\Gamma\}\.\(102\)
As a result,

D\\displaystyle D=D1\+D2\\displaystyle=D\_\{1\}\+D\_\{2\}≤\(1−μ​ηt\+1\)​𝖤​‖𝒘t−𝒘∗‖2\+1M​∑k=1M𝖤​‖𝒘t−𝒘tk‖2\+\\displaystyle\\leq\(1\-\{\\mu\}\{\\eta\}\_\{t\+1\}\)\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\+\\frac\{1\}\{M\}\\sum\_\{k=1\}^\{M\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|^\{2\}\+4​L​ηt\+12​Γ\+ηt\+12M2​∑k=1Mσk2\.\\displaystyle\\quad 4L\{\\eta\}\_\{t\+1\}^\{2\}\{\\Gamma\}\+\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\{\\sigma\}\_\{k\}^\{2\}\.\(103\)
And

𝖤​‖𝒘t−𝒘tk‖2\\displaystyle\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\}^\{k\}\\right\\\|^\{2\}=𝖤​‖\(𝒘t−𝒘t−1\)−\(𝒘tk−𝒘t−1\)‖2\\displaystyle=\\mathsf\{E\}\\left\\\|\\left\(\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}\_\{t\-1\}\\right\)\-\\left\(\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}\_\{t\-1\}\\right\)\\right\\\|^\{2\}≤\(a\)𝖤​‖𝒘tk−𝒘t−1‖2\\displaystyle\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{\\leq\}\}\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}\_\{t\-1\}\\right\\\|^\{2\}≤ηt2​G2\\displaystyle\\leq\{\\eta\}\_\{t\}^\{2\}G^\{2\}≤\(b\)4​ηt\+12​G2\.\\displaystyle\\stackrel\{\{\\scriptstyle\(b\)\}\}\{\{\\leq\}\}4\{\\eta\}\_\{t\+1\}^\{2\}G^\{2\}\.\(104\)where \(a\) is because𝖤​‖X−𝖤⁡\[X\]‖2≤𝖤​‖X‖2\\mathsf\{E\}\\left\\\|X\-\\mathsf\{E\}\\left\[X\\right\]\\right\\\|^\{2\}\\leq\\mathsf\{E\}\\left\\\|X\\right\\\|^\{2\}, in whichX=𝒘tk−𝒘t−1X=\\boldsymbol\{w\}\_\{t\}^\{k\}\-\\boldsymbol\{w\}\_\{t\-1\}and \(b\) is because the step sizes\{ηt\}\\\{\{\\eta\}\_\{t\}\\\}are non\-increasing and satisfyηt≤2​ηt\+1\{\\eta\}\_\{t\}\\leq 2\{\\eta\}\_\{t\+1\}\. Substituting \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex99)\) into \([IV\-D](https://arxiv.org/html/2608.13961#S4.Ex97)\), we obtain that

D≤\\displaystyle D\\leq\(1−μ​ηt\+1\)​𝖤​‖𝒘t−𝒘∗‖2\+4​ηt\+12​G2\+4​L​ηt\+12​Γ\\displaystyle\(1\-\{\\mu\}\{\\eta\}\_\{t\+1\}\)\\mathsf\{E\}\\left\\\|\\boldsymbol\{w\}\_\{t\}\-\\boldsymbol\{w\}^\{\*\}\\right\\\|^\{2\}\+4\{\\eta\}\_\{t\+1\}^\{2\}G^\{2\}\+4L\{\\eta\}\_\{t\+1\}^\{2\}\{\\Gamma\}\+ηt\+12M2∑k=1Mσk2\.\\displaystyle\+\\frac\{\{\\eta\}\_\{t\+1\}^\{2\}\}\{M^\{2\}\}\\sum\_\{k=1\}^\{M\}\{\\sigma\}\_\{k\}^\{2\}\.\(105\)

## VAllocation of channels uses and quantization bits

The convergence analysis in the previous section tacitly assumes that the number of channelsNNand the number of quantization bitsnnremain unchanged through all iterations\. From \([32](https://arxiv.org/html/2608.13961#S4.E32)\), we can see that asttincreases, the term\[∏j=t\+1T\(1−μ​ηj\)\]\\left\[\\prod\_\{j=t\+1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]becomes larger, which means that the termηt2​H\{\\eta\}\_\{t\}^\{2\}Hcontributes more in the later stage than that in the early stage\. Intuitively, we should letnnandNNvary along the iterations such thatHtH\_\{t\}is a function ofttand decreases asttincreases\.

In this section, we will optimize the upper bound of the convergence performance, i\.e\., the right hand side of \([32](https://arxiv.org/html/2608.13961#S4.E32)\), over the number of channelsNNand the number of quantization bitsnn\. From the right hand side of \([32](https://arxiv.org/html/2608.13961#S4.E32)\) and the definition ofHHin \([33](https://arxiv.org/html/2608.13961#S4.E33)\), we note that the number of channelsNNand the number of quantization bitsnnonly affect the transmission lossd​Δ2​∑i=1n4i−1​Z​\(WN\(i\)\)d\{\\Delta\}^\{2\}\\sum\_\{i=1\}^\{n\}4^\{i\-1\}Z\\left\(W\_\{N\}^\{\(i\)\}\\right\)and the quantization lossd6​Δ2\\frac\{d\}\{6\}\{\\Delta\}^\{2\}, whereΔ=Bm​a​x−Bm​i​n2n−1\\Delta=\\frac\{B\_\{max\}\-B\_\{min\}\}\{2^\{n\}\-1\}\. For the simplicity of the optimization problem, we use a constant step size hereη\\eta\. As a result, the loss function in the optimization can be simplified toη2​∑t=1T\(1−μ​η\)T−t​Ht\{\\eta\}^\{2\}\\sum\_\{t=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\)^\{T\-t\}\{H\}\_\{t\}\. By rearranging the loss function and removing the constant term inHt\{H\}\_\{t\}, we obtain the following optimization problem

P1:minNt,nt\\displaystyle P1:\\quad\\min\_\{N\_\{t\},n\_\{t\}\}\\quad∑t=1T\(1−μ​η\)T−t​16\+∑i=1nt4i−1​Z​\(WNt\(i\)\)\(2nt−1\)2\\displaystyle\\sum\_\{t=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\)^\{T\-t\}\\frac\{\\frac\{1\}\{6\}\+\\sum\_\{i=1\}^\{n\_\{t\}\}4^\{i\-1\}Z\\left\(W\_\{N\_\{t\}\}^\{\(i\)\}\\right\)\}\{\(2^\{n\_\{t\}\}\-1\)^\{2\}\}\(106\)s\.t\.nt∈\{1,⋯,Nt\},\\displaystyle n\_\{t\}\\in\\\{1,\\cdots,N\_\{t\}\\\},\(107\)∑t=1TNt≤T​Na​v​e,\\displaystyle\\sum\_\{t=1\}^\{T\}N\_\{t\}\\leq T\{N\}\_\{ave\},\(108\)Nt∈\{16,32,64,128\},\\displaystyle N\_\{t\}\\in\\\{16,32,64,128\\\},\(109\)where constraint \([108](https://arxiv.org/html/2608.13961#S5.E108)\) means the total number of channels used in the learning procedure is fixed andNa​v​e\{N\}\_\{ave\}is a constant which stands for the average number of channel uses in each iteration\. Constraint \([109](https://arxiv.org/html/2608.13961#S5.E109)\) limits the number of channel uses in each iteration to the power of22due to the structure of polar codes\.

Before we proceed with the optimization, we note that the Bhattacharyya parameterZ⁡\(WN\(l\)\)Z\(W\_\{N\}^\{\(l\)\}\)for BEC, which appears in the above loss function, can only be computed analytically in a recursive way\. The absence of explicit closed\-form expression hinders the further optimization\. Therefore, we use Gaussian cumulative distribution function \(CDF\) to approximate the Bhattacharyya parameters in BEC channel\. We define

γ⁡\(x\)=0\.5∗erf⁡\(x−μσ\)\+0\.5\.\\displaystyle\\gamma\(x\)=0\.5\\ast\\operatorname\{erf\}\(\\frac\{x\-\\mu\}\{\\sigma\}\)\+0\.5\.\(110\)where

erf⁡\(x\)\\displaystyle\\operatorname\{erf\}\(x\)=2π​∫0xe−t2​𝑑t\\displaystyle=\\frac\{2\}\{\\sqrt\{\\pi\}\}\\int\_\{0\}^\{x\}e^\{\-t^\{2\}\}dt\(111\)withμ\\muandσ\\sigmabeing mean and variance of the Gaussian distribution respectively\. We demonstrate an illustrative example of fitting the Bhattacharyya parameters of 64 BECs byγ⁡\(x\)\\gamma\(x\)function in Fig\.[3](https://arxiv.org/html/2608.13961#S5.F3)with the erasure probability being 0\.2, 0\.4, 0\.6 and 0\.8 respectively\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/Fitting_result_02.png)\(a\)erasure probability 0\.2
![Refer to caption](https://arxiv.org/html/2608.13961v1/Fitting_result_04.png)\(b\)erasure probability 0\.4
![Refer to caption](https://arxiv.org/html/2608.13961v1/Fitting_result_06.png)\(c\)erasure probability 0\.6
![Refer to caption](https://arxiv.org/html/2608.13961v1/Fitting_result_08.png)\(d\)erasure probability 0\.8

Fig\. 3:Fitting results of 64 BECs\.Further, we approximate both parametersμ\\muandσ\\sigmaas linear functions of the block lengthNNas

μ=aϵ​N\+bϵ\\displaystyle\\mu=a\_\{\\epsilon\}N\+b\_\{\\epsilon\}\(112\)σ=cϵ​N\+dϵ\\displaystyle\\sigma=c\_\{\\epsilon\}N\+d\_\{\\epsilon\}\(113\)where the parametersaϵ,bϵ,cϵ,dϵa\_\{\\epsilon\},b\_\{\\epsilon\},c\_\{\\epsilon\},d\_\{\\epsilon\}depend on the erasure probabilityϵ\\epsilon\. We demonstrate the fitting performance ofμ\\muandσ\\sigmaforϵ=0\.6\\epsilon=0\.6in Fig\.[4](https://arxiv.org/html/2608.13961#S5.F4)\. The overall fitting results of ofμ\\muandσ\\sigmaby parametersaϵ,bϵ,cϵ,dϵa\_\{\\epsilon\},b\_\{\\epsilon\},c\_\{\\epsilon\},d\_\{\\epsilon\}under different erasure probabilities are listed in Table[I](https://arxiv.org/html/2608.13961#S5.T1)\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/N_mu.png)\(a\)NNandμ\\mu\.
![Refer to caption](https://arxiv.org/html/2608.13961v1/N_sigma.png)\(b\)NNandσ\\sigma\.

Fig\. 4:The fitted function ofNNvsμ\\muandNNvsσ\\sigmawithϵ=0\.6\\epsilon=0\.6\.TABLE I:Fitting resultsaϵ,bϵ,cϵ,dϵa\_\{\\epsilon\},b\_\{\\epsilon\},c\_\{\\epsilon\},d\_\{\\epsilon\}under differentϵ\\epsilon\.Further numerical fitting performance of the Bhattacharyya parametersfor different block length and erasure probabilities, in terms of sum of Squares due to Error\(SSE\), R\-squared\(R2R^\{2\}\), Adjusted R\-squared\(A​R2AR^\{2\}\), Root Mean Squared Error\(RMSE\), can be found in Table[II](https://arxiv.org/html/2608.13961#S5.T2)\. In all the above cases, we can see that the Bhattacharyya parameters can be approximated by the Gaussian distribution functionγ⁡\(x\)\\gamma\(x\)very accurately\.

TABLE II:Fitting performance for different number of channel uses\.![Refer to caption](https://arxiv.org/html/2608.13961v1/NMSE_32.png)Fig\. 5:The normalized MSE with N=32\.### V\-AOptimization of number of quantization bitsntn\_\{t\}

For the fixedNtN\_\{t\}, when number of quantization bitsntn\_\{t\}increases, the quantization loss decreases\. At the same time, the channels used to transmit the extra quantization bits become less reliable\. The conflicting effects ofntn\_\{t\}is illustrated in term of normalized mean square error \(NMSE\), which is defined in the following equation, in Fig\.[5](https://arxiv.org/html/2608.13961#S5.F5)\.

N​M​S​E=‖Q⁡\(𝒈t\+1,jk\)−Q~​\(𝒈t\+1,jk\)‖2‖Q⁡\(𝒈t\+1,jk\)‖2\.\\displaystyle NMSE=\\frac\{\\left\\\|Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\-\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\\\|^\{2\}\}\{\\left\\\|Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)\\right\\\|^\{2\}\}\.\(114\)whereQ⁡\(𝒈t\+1,jk\)Q\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)is the quantized gradient prior to the channel coding andQ~​\(𝒈t\+1,jk\)\\tilde\{Q\}\(\\boldsymbol\{g\}\_\{t\+1,j\}^\{k\}\)is the reconstruction of the gradient based on the channel coding output\. We use NMSE to quantitatively evaluate the effect ofnn, the number of quantization bits, for a given polar code\. We note that when the erasure probability of the channels is low, a largernnyields lower NMSE and when erasure probability increases, a smallernngives a better NMSE\.

Based on the above analysis, we will take a sequential approach to the optimization problemP​1P1\. We fix the parameterNtN\_\{t\}and find an optimalntn\_\{t\}for the problemP​1P1\.

For example, assumeNt=32N\_\{t\}=32andϵ=0\.6\\epsilon=0\.6, and the optimization problemP​1P1can be simplified as

minnt\\displaystyle\\min\_\{n\_\{t\}\}\\quad16\+∑i=1nt4i−1​\[12​erf⁡\(i−13\.226\.456\)\+12\]\(2nt−1\)2\\displaystyle\\frac\{\\frac\{1\}\{6\}\+\\sum\_\{i=1\}^\{n\_\{t\}\}4^\{i\-1\}\[\\frac\{1\}\{2\}\\operatorname\{erf\}\(\\frac\{i\-13\.22\}\{6\.456\}\)\+\\frac\{1\}\{2\}\]\}\{\(2^\{n\_\{t\}\}\-1\)^\{2\}\}\(115\)s\.t\.nt∈\{1,⋯,N\}\.\\displaystyle n\_\{t\}\\in\\\{1,\\cdots,N\\\}\.\(116\)
![Refer to caption](https://arxiv.org/html/2608.13961v1/optimization.png)Fig\. 6:The curve of objective function\.The loss function in the above example is depicted in Fig\.[6](https://arxiv.org/html/2608.13961#S5.F6)\. We note that the objective function first decreases and then increases withntn\_\{t\}increasing, and the optimalntn\_\{t\}is44\. A further inspection on the curve of the objective function shows that the value of objective function is about the same if we selectntn\_\{t\}from\{3,4,5\}\\\{3,4,5\\\}, which we deem as “good” choices\. If we takeϵ=0\.1,…,0\.8\\epsilon=0\.1,\\dots,0\.8and takeNNfor16,32,64,12816,32,64,128in the optimization problem\. The corresponding curves have the similar shapes as Fig\.[6](https://arxiv.org/html/2608.13961#S5.F6)\. Meanwhile whenNNincreases, the size of the set of “good” choices increases\. We conclude the above observations in Table[III](https://arxiv.org/html/2608.13961#S5.T3)for various erasure probabilityϵ\\epsilonand block lengthNN\.

TABLE III:“good” choices ofnnunder differentNNandϵ\\epsilonFrom the above observation summarized in Table[III](https://arxiv.org/html/2608.13961#S5.T3), we note that given the erasure probabilityϵ\\epsilon, we can find a constantnnwhich is “good” for all the block lengthN=16,32,64,128N=16,32,64,128\. As the erasure probabilityϵ\\epsilonincreases, the number of “good” choices ofnndecreases, which makes the constant harder to find\. To an extreme, sayϵ=0\.8\\epsilon=0\.8, although there is no constantnn“good” for allN=16,32,64,128N=16,32,64,128, we can taken=3n=3as a compromise\.

So the conclusion is that the parameterntn\_\{t\}is only a function ofϵ\\epsilonand it does not depend on either the block lengthNNor the iterationtt\. In the next subsubsection, we will optimizeNtN\_\{t\}by keepingnnfixed\.

### V\-BOptimization of the block lengthNtN\_\{t\}

According to the analysis in the previous subsection, for a given erasure probabilityϵ\\epsilon, we fix the parameternn, and the optimization problemP1P\_\{1\}can be simplified as

P​2min\{Nt\}\\displaystyle P2\\quad\\min\_\{\\\{N\_\{t\}\\\}\}\\quad∑t=1T\(1−μ​η\)T−t​\[16\+∑i=1n4i−1​Z​\(WNt\(i\)\)\]\\displaystyle\\sum\_\{t=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\)^\{T\-t\}\\left\[\\frac\{1\}\{6\}\+\\sum\_\{i=1\}^\{n\}4^\{i\-1\}Z\\left\(W\_\{N\_\{t\}\}^\{\(i\)\}\\right\)\\right\]\(117\)s\.t\.∑t=1TNt≤T​Na​v​e,\\displaystyle\\sum\_\{t=1\}^\{T\}N\_\{t\}\\leq T\{N\}\_\{ave\},\(118\)Nt∈\{16,32,64,128\}\.\\displaystyle N\_\{t\}\\in\\\{16,32,64,128\\\}\.\(119\)
By substituting the approximation ofZ⁡\(WNt\(i\)\)Z\\left\(W\_\{N\_\{t\}\}^\{\(i\)\}\\right\), the optimization problem is simplified as follows

P​3min\{Pt\}\\displaystyle P3\\quad\\min\_\{\\\{P\_\{t\}\\\}\}\\quad∑t=1T\(1−μ​η\)T−t​\[16\+∑i=1n4i−1​\[12​G​\(Pt\)\+12\]\]\\displaystyle\\sum\_\{t=1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\)^\{T\-t\}\\left\[\\frac\{1\}\{6\}\+\\sum\_\{i=1\}^\{n\}4^\{i\-1\}\\left\[\\frac\{1\}\{2\}G\(P\_\{t\}\)\+\\frac\{1\}\{2\}\\right\]\\right\]\(120\)s\.t\.∑t=1T2Pt≤T​Na​v​e,\\displaystyle\\sum\_\{t=1\}^\{T\}\{2\}^\{P\_\{t\}\}\\leq T\{N\}\_\{ave\},\(121\)Pt∈ℤ,4≤Pt≤7,∀t∈\{1,…,T\},\\displaystyle P\_\{t\}\\in\\mathbb\{Z\},\\quad 4\\leq P\_\{t\}\\leq 7,\\quad\\forall t\\in\\\{1,\\ldots,T\\\},\(122\)G⁡\(Nt\)=erf⁡\(i−aϵ∗2Pt−bϵcϵ∗2Pt\+dϵ\)\\displaystyle G\(N\_\{t\}\)=\\operatorname\{erf\}\\left\(\\frac\{i\-a\_\{\\epsilon\}\\ast\{2\}^\{P\_\{t\}\}\-b\_\{\\epsilon\}\}\{c\_\{\\epsilon\}\\ast\{2\}^\{P\_\{t\}\}\+d\_\{\\epsilon\}\}\\right\)\(123\)
The above optimization problem is a mixed\-integer nonlinear programming \(MINLP\) problem, which can be solved by particle swarm optimization \(PSO\) algorithm\.

To show the advantage of the proposed solution, we compare it to the benchmark scheme which uses a constant block length through all iteration\. We set the number of iterationsT=35T=35\.

Benchmark:The benchmark scheme uses a fixed block lengthN=16N=16\.

Proposed:The proposed scheme uses the solution of the optimization problem \([106](https://arxiv.org/html/2608.13961#S5.E106)\) where we set the average block lengthNa​v​e=16N\_\{ave\}=16\.

We use the second term in the upper bound of the convergence analysisL2​∑t=1T\[∏j=t\+1T\(1−μ​ηj\)\]​ηt2​Ht\\frac\{L\}\{2\}\\sum\_\{t=1\}^\{T\}\\left\[\\prod\_\{j=t\+1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]\{\\eta\}\_\{t\}^\{2\}H\_\{t\}in Theorem[1](https://arxiv.org/html/2608.13961#ThmTheo1)as a criterion to compare the performance of the two schemes\. To further demonstrate the difference between two schemes, we plot the the accumulations in the above criterion iteration by iteration asy=L2​∑t=1t′\[∏j=t\+1T\(1−μ​ηj\)\]​ηt2​Hty=\\frac\{L\}\{2\}\\sum\_\{t=1\}^\{t^\{\\prime\}\}\\left\[\\prod\_\{j=t\+1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]\{\\eta\}\_\{t\}^\{2\}H\_\{t\}fort′∈\{1,⋯,T\}t^\{\\prime\}\\in\\\{1,\\cdots,T\\\}for two schemes in Fig\.[7](https://arxiv.org/html/2608.13961#S5.F7)whereϵ=0\.6\\epsilon=0\.6\. The superiority of the proposed scheme becomes apparent in the later stage of iteration\. This is intuitively expected, as the term\[∏j=t\+1T\(1−μ​ηj\)\]​ηt2​Ht\\left\[\\prod\_\{j=t\+1\}^\{T\}\(1\-\{\\mu\}\{\\eta\}\_\{j\}\)\\right\]\{\\eta\}\_\{t\}^\{2\}H\_\{t\}undergoes weaker suppression in this stage, which naturally motivates our proposed scheme to allocate additional resources to the this stage\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/compare_upper_bound.png)Fig\. 7:The convergence upper bound of the proposed scheme and benchmark scheme\.In another demonstration, we use the number of channel uses instead of the number of iterations as the horizontal coordinate, to better reflect the convergence speed in term of time, and convergence upper bound as the vertical coordinate\. As shown in Fig\.[8](https://arxiv.org/html/2608.13961#S5.F8), during35×1635\\times 16channel uses, the proposed channel allocation scheme achieves the convergence better and faster than the constant block length benchmark\. The reason is that the proposed channel allocation scheme use shorter block length for the early stage of the iterations, which yields a faster convergence, a phenomenon not obvious in Fig\.[7](https://arxiv.org/html/2608.13961#S5.F7)where number of iterations is used as horizontal coordinate, and a longer block length for the later stage of the iterations, which gives a better convergence especially with large erasure probabilityϵ\\epsilon\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/Compare_FL_02.png)\(a\)erasure probability 0\.2
![Refer to caption](https://arxiv.org/html/2608.13961v1/Compare_FL_06.png)\(b\)erasure probability 0\.6
![Refer to caption](https://arxiv.org/html/2608.13961v1/Compare_FL_07.png)\(c\)erasure probability 0\.7
![Refer to caption](https://arxiv.org/html/2608.13961v1/Compare_FL_08.png)\(d\)erasure probability 0\.8

Fig\. 8:The test accuracy of different schemes\.

## VINumerical Results

In this section, we consider an image classification task in the following experiments\. All programs are implemented in python 3\.8 and Pytorch\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP01.png)\(a\)erasure probability 0\.1
![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP02.png)\(b\)erasure probability 0\.2
![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP03.png)\(c\)erasure probability 0\.3
![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP04.png)\(d\)erasure probability 0\.4
![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP05.png)\(e\)erasure probability 0\.5
![Refer to caption](https://arxiv.org/html/2608.13961v1/TAEP08.png)\(f\)erasure probability 0\.8

Fig\. 9:The test accuracy of different schemesIn the experiments, we use the well\-known MNIST data set for the training of the convolutional neural networks \(CNN\)\. The MNIST data set has7000070000gray images, including6000060000samples for training and1000010000samples for testing\. These images with the same size of28×2828\\times 28can be categorized into ten classes, each containing a handwritten digit from00to99\. The convolution neural network we used for experiments comprises two5×55\\times 5convolution layers\. Specifically, the first layer has1010output channels, while the second layer contains2020output channels, and each layer is connected with a2×22\\times 2maximum pooling layer and then the ReLu activation function\. Next is a full connection layer consisting of 50 output channels with the ReLu activation function\. Then after a dropout layer is another full connection layer consisting of1010output classes\. The batch size is set to100100and the learning optimizer uses SGD\. Cross\-entropy is used as the loss function and the learning step size is set to be0\.0050\.005\.

We assume thatM=20M=20clients are connected to one server\. Each client carries30003000samples\. The data is independent and identically distributed \(IID\) and the training data is shuffled and then randomly assigned to each user\. For each global iteration,2020percent of the2020clients are selected to participate in this learning iteration and the whole learning process lasts forT=40T=40global iterations\. To demonstrate the advantage of our proposed scheme, we compare the performance of the proposed scheme with the following benchmarks:

Uncoded transmission:This scheme uses no channel coding\. The data are quantized into3232bits and then sent directly into3232uses of the binary erasure channels\.

LDPC coded transmission:This scheme quantizes the data into55bits and then uses an EEP LDPC codes with a constant code lengthN=32N=32\.

Polar coded transmission with constant block length:This scheme quantizes the data into55bits and then uses a UEP polar code with a constant code lengthN=32N=32, as proposed in Section[III](https://arxiv.org/html/2608.13961#S3)\.

Polar coded transmission with variable block length:This scheme quantizes the data intonnbits, which is a function of the erasure probabilityϵ\\epsilon, more specifically,n=5n=5for0≤ϵ≤0\.40\\leq\\epsilon\\leq 0\.4, taken=4n=4for0\.5≤ϵ≤0\.60\.5\\leq\\epsilon\\leq 0\.6and taken=3n=3forϵ≥0\.7\\epsilon\\geq 0\.7\. Then it uses a UEP polar code with a variable block length\{Nt\}\\\{N\_\{t\}\\\}according to the solution of the optimization problemP​3P3in Section[V](https://arxiv.org/html/2608.13961#S5)\. The performance criterion in the experiments is the accuracy, the percentage of correctness when applying the converging cnn or the converged cnn on the testing data\.

We demonstrate the convergence performances, i\.e\., accuracy on the testing data for the converging cnn, of four schemes under different erase probabilities in Fig\.[9](https://arxiv.org/html/2608.13961#S6.F9)\. We note that the proposed polar code based schemes, with constant or variable block length, have a faster convergence speed and a higher accuracy than the two benchmark schemes, i\.e\., uncoded and LDPC schemes, especially when the erasure probability is large\.

Fig\.[10](https://arxiv.org/html/2608.13961#S6.F10)presents the test accuracy of the converged CNN, offering a more complete performance comparison\. The proposed polar code based schemes, whether employing constant or variable block lengths, consistently achieve substantial gains over both the uncoded and LDPC benchmarks\. This marked improvement is largely attributable to the unequal error protection \(UEP\) property of the polar code, as detailed in Section[III](https://arxiv.org/html/2608.13961#S3)\. Compared with the constant\-length configuration, the variable\-length strategy introduced in Section[V](https://arxiv.org/html/2608.13961#S5)yields only a modest additional gain, suggesting that the optimization over block lengths may be safely omitted when its computational overhead is prohibitively expensive\. Furthermore, consistent with the trend observed in Fig\.[9](https://arxiv.org/html/2608.13961#S6.F9), the results in Fig\.[10](https://arxiv.org/html/2608.13961#S6.F10)demonstrate that the performance advantage of the proposed scheme over the benchmarks becomes increasingly pronounced as the channel erasure probability increases\.

![Refer to caption](https://arxiv.org/html/2608.13961v1/CA.png)Fig\. 10:The test accuracy after convergence of different schemes under different erase probabilities\.
## VIIConclusion

In this paper, we propose a polar code based federated learning \(FL\) scheme that leverages the unequal error protection \(UEP\) property of polar codes under finite block lengths to protect the quantization bits of the local model according to their relative significance\. We further conduct a convergence analysis of the proposed scheme and optimize the upper bound on the convergence gap over the number of quantization bits and the polar code block length across all training iterations\. Experimental results demonstrate that the proposed polar code based schemes, whether with constant or variable block lengths, achieve significant performance gains over both uncoded and LDPC\-based benchmarks\.

## References

- \[1\]G\. Zhu, D\. Liu, Y\. Du, C\. You, J\. Zhang, and K\. Huang, “Toward an intelligent edge: Wireless communication meets machine learning,”*IEEE communications magazine*, vol\. 58, no\. 1, pp\. 19–25, 2020\.
- \[2\]J\. Konečnỳ, H\. B\. McMahan, D\. Ramage, and P\. Richtárik, “Federated optimization: Distributed machine learning for on\-device intelligence,”*arXiv preprint arXiv:1610\.02527*, 2016\.
- \[3\]B\. McMahan, E\. Moore, D\. Ramage, S\. Hampson, and B\. A\. y Arcas, “Communication\-efficient learning of deep networks from decentralized data,” in*Artificial intelligence and statistics*\. Pmlr, 2017, pp\. 1273–1282\.
- \[4\]Y\. Oh, N\. Lee, Y\.\-S\. Jeon, and H\. V\. Poor, “Communication\-efficient federated learning via quantized compressed sensing,”*IEEE Transactions on Wireless Communications*, vol\. 22, no\. 2, pp\. 1087–1100, 2022\.
- \[5\]M\. Zhang, G\. Zhu, S\. Wang, J\. Jiang, Q\. Liao, C\. Zhong, and S\. Cui, “Communication\-efficient federated edge learning via optimal probabilistic device scheduling,”*IEEE Transactions on Wireless Communications*, vol\. 21, no\. 10, pp\. 8536–8551, 2022\.
- \[6\]Z\. Ni, Z\. Zhang, N\. C\. Luong, D\. Niyato, D\. I\. Kim, and S\. Feng, “Joint client scheduling and quantization optimization in energy harvesting\-enabled federated learning networks,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 8, pp\. 9566–9582, 2024\.
- \[7\]Z\. Chen, W\. Yi, and A\. Nallanathan, “Exploring representativity in device scheduling for wireless federated learning,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 1, pp\. 720–735, 2024\.
- \[8\]M\. Ma, V\. W\. Wong, and R\. Schober, “Channel\-aware joint aoi and diversity optimization for client scheduling in federated learning with non\-iid datasets,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 6, pp\. 6295–6311, 2024\.
- \[9\]J\. Yang, Y\. Liu, F\. Chen, W\. Chen, and C\. Li, “Asynchronous wireless federated learning with probabilistic client selection,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 7, pp\. 7144–7158, 2024\.
- \[10\]K\. Wang, Z\. Ding, D\. K\. C\. So, and Z\. Ding, “Age\-of\-information minimization in federated learning based networks with non\-iid dataset,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 8, pp\. 8939–8953, 2024\.
- \[11\]Z\. Chen, W\. Yi, H\. Shin, and A\. Nallanathan, “Adaptive model pruning for communication and computation efficient wireless federated learning,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 7, pp\. 7582–7598, 2024\.
- \[12\]B\. Luo, W\. Xiao, S\. Wang, J\. Huang, and L\. Tassiulas, “Adaptive heterogeneous client sampling for federated learning over wireless networks,”*IEEE Transactions on Mobile Computing*, vol\. 23, no\. 10, pp\. 9663–9677, 2024\.
- \[13\]Z\. Yan and D\. Li, “Performance analysis for resource constrained decentralized federated learning over wireless networks,”*IEEE Transactions on Communications*, vol\. 72, no\. 7, pp\. 4084–4100, 2024\.
- \[14\]B\. Wu, F\. Fang, and X\. Wang, “Joint age\-based client selection and resource allocation for communication\-efficient federated learning over noma networks,”*IEEE Transactions on Communications*, vol\. 72, no\. 1, pp\. 179–192, 2024\.
- \[15\]S\. Mao, L\. Liu, N\. Zhang, J\. Hu, K\. Yang, F\. R\. Yu, and V\. C\. M\. Leung, “Intelligent reflecting surface\-assisted low\-latency federated learning over wireless networks,”*IEEE Internet of Things Journal*, vol\. 10, no\. 2, pp\. 1223–1235, 2023\.
- \[16\]P\. Zheng, Y\. Zhu, Y\. Hu, Z\. Zhang, and A\. Schmeink, “Federated learning in heterogeneous networks with unreliable communication,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 4, pp\. 3823–3838, 2024\.
- \[17\]M\. Chen, Z\. Yang, W\. Saad, C\. Yin, H\. V\. Poor, and S\. Cui, “A joint learning and communications framework for federated learning over wireless networks,”*IEEE Transactions on Wireless Communications*, no\. 1, 2021\.
- \[18\]Y\. Wang, Y\. Xu, Q\. Shi, and T\. H\. Chang, “Quantized federated learning under transmission delay and outage constraints,”*Selected Areas in Communications, IEEE Journal on \(J\-SAC\)*, vol\. 40, no\. 1, 2022\.
- \[19\]H\. Chen, S\. Huang, D\. Zhang, M\. Xiao, M\. Skoglund, and H\. V\. Poor, “Federated learning over wireless iot networks with optimized communication and resources,”*IEEE Internet of Things Journal*, vol\. 9, no\. 17, p\. 14, 2022\.
- \[20\]LiBoning, PerazzoneJake, SwamiAnanthram, and SegarraSantiago, “Learning to transmit with provable guarantees in wireless federated learning,”*IEEE Transactions on Wireless Communications*, 2024\.
- \[21\]X\. Hou, J\. Wang, C\. Jiang, Z\. Meng, J\. Chen, and Y\. Ren, “Efficient federated learning for metaverse via dynamic user selection, gradient quantization and resource allocation,”*IEEE Journal on Selected Areas in Communications*, vol\. 42, no\. 4, pp\. 850–866, 2024\.
- \[22\]H\. Li, R\. Wang, W\. Zhang, and J\. Wu, “One bit aggregation for federated edge learning with reconfigurable intelligent surface: Analysis and optimization,”*IEEE Transactions on Wireless Communications*, vol\. 22, no\. 2, pp\. 872–888, 2023\.
- \[23\]H\. H\. Yang, Z\. Liu, T\. Q\. S\. Quek, and H\. V\. Poor, “Scheduling policies for federated learning in wireless networks,”*IEEE Transactions on Communications*, vol\. PP, no\. 99, pp\. 1–1, 2019\.
- \[24\]T\. M\. Cover and J\. A\. Thomas, “Elements of information theory second edition solutions to problems,”*Internet Access*, pp\. 19–20, 2006\.
- \[25\]M\. Salehi and E\. Hossain, “Federated learning in unreliable and resource\-constrained cellular wireless networks,”*IEEE Transactions on Communications*, no\. 8, 2021\.
- \[26\]S\. Zheng, C\. Shen, and X\. Chen, “Design and analysis of uplink and downlink communications for federated learning,”*IEEE Journal on Selected Areas in Communications*, vol\. 39, no\. 7, pp\. 2150–2167, 2020\.
- \[27\]L\. Chen, W\. Liu, Y\. Chen, and W\. Wang, “Communication\-efficient design for quantized decentralized federated learning,”*IEEE Transactions on Signal Processing*, vol\. 72, pp\. 1175–1188, 2024\.
- \[28\]E\. Arikan, “Channel polarization: A method for constructing capacity\-achieving codes for symmetric binary\-input memoryless channels,”*IEEE Transactions on Information Theory*, vol\. 55, no\. 7, pp\. 3051–3073, 2009\.
- \[29\]P\. S\. Bouzinis, P\. D\. Diamantoulakis, and G\. K\. Karagiannidis, “Wireless quantized federated learning: A joint computation and communication design,”*IEEE Transactions on Communications*, vol\. 71, no\. 5, pp\. 2756–2770, 2023\.
- \[30\]C\. Xu, J\. Li, Y\. Liu, Y\. Ling, and M\. Wen, “Accelerating split federated learning over wireless communication networks,”*IEEE Transactions on Wireless Communications*, vol\. 23, no\. 6, pp\. 5587–5599, 2024\.
- \[31\]H\. H\. Yang, Z\. Chen, and T\. Q\. S\. Quek, “Unleashing edgeless federated learning with analog transmissions,”*IEEE Transactions on Signal Processing*, vol\. 72, pp\. 774–791, 2024\.
- \[32\]T\. Sery and K\. Cohen, “On analog gradient descent learning over multiple access fading channels,”*IEEE Transactions on Signal Processing*, vol\. 68, pp\. 2897–2911, 2020\.
- \[33\]M\. M\. Amiri, D\. Gunduz, S\. R\. Kulkarni, and H\. V\. Poor, “Federated learning with quantized global model updates,”*arXiv preprint arXiv:2006\.10672*, 2020\.